Optical modules and method of precisely assembling same

ABSTRACT

Optical modules and a method of precisely assembling the modules are provided. The method includes the step of providing a set of optical components including a plurality of beam shaping components for converting at least one initial shape in cross section of a laser beam to at least one desired shape in cross section and an optical mount for supporting the set of optical components. The method further includes holding and locating the optical mount relative to a reference axis. The method still further includes holding the optical components in position relative to the optical mount during the step of holding and locating wherein the positions of the beam shaping components are initial positions. The method includes directing a laser beam having the at least one initial shape at the set of optical components during the step of holding the optical components. The method further includes adjusting the initial positions of the beam shaping components to obtain final positions of the beam shaping components relative to the optical mount during the step of directing until the laser beam has the at least one desired shape.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related to the following commonly-owned U.S. patentapplications which were filed on Oct. 23, 2007:

-   1) Method and System for Optically Inspecting Parts (U.S. Ser. No.    11/977,117);-   2) Method for Estimating Thread Parameters of a Part (U.S. Ser. No.    11/977,097);-   3) Method and Inspection Head Apparatus for Optically Measuring    Geometric Dimensions of a Part (U.S. Ser. No. 11/977,010);-   4) Apparatus for Quickly Retaining and Releasing Parts to be    Optically Measured (U.S. Ser. No. 11/977,091);-   5) Calibration Device for Use in an Optical Part Measuring System    (U.S. Ser. No. 11/977,114); and-   6) Method and System for Generating Calibration Data For Use In    Calibrating A Part Inspection System (U.S. Ser. No. 11/975,977).

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to optical modules and methods of preciselyassembling same.

2. Background Art

Traditional manual, gauging devices and techniques have been replaced tosome extent by automatic inspection methods and systems. However, suchautomatic inspection methods and systems still have a number ofshortcomings associated with them.

WO 2005/022076 discloses a plurality of light line generators (72) whichgenerate associated beams of light (26) that intersect a part (14) to beinspected. Each beam of light (26) illuminates at least one side of thepart (14) with a line of light occluded by the part (14), and at leastthree light responsive sensors (104) provide for generating a signal(24) responsive to an occlusion of a corresponding line of light on acorresponding side of at least one side of the part (14). Each of thelight responsive sensors is responsive to an occlusion at a differentazimuthal location. A processor (28) analyzes the signals (24) inrelation to a measure of relative location of the part (14) from amotion (18) or position sensor. The part (14) may be released from aclamp (52) to drop through the beams of light (26), or the beams oflight (26) may be moved relative to the part (14).

U.S. Pat. No. 6,313,948 discloses an optical beam shaper for productionof a uniform sheet of light for use in a parts inspection system havinga light source including a coherent light generator, a diffractive beamshaper, and lens elements.

U.S. Pat. No. 6,285,031 discloses an inspection system for evaluatingrotationally asymmetric workpieces for conformance to configurationcriteria. The system has a track for causing the workpieces to translatethrough a test section. The test section includes a plurality ofelectromagnetic energy sources. The plurality of electromagnetic energysources are oriented with respect to the track such that the workpiecesocclude the plurality of electromagnetic energy sources upon passingthrough the test section. The test section further has electromagneticenergy detectors for receiving the electromagnetic energy to provideoutput signals related to the intensity of the occluded electromagneticenergy incident on the electromagnetic energy detectors, and a signalprocessor for receiving and processing the output signals.

U.S. Pat. No. 6,252,661 discloses an inspection system for evaluatingworkpieces for conformance to configuration criteria. The systemincludes a track for causing workpieces to translate through a testsection. The test section includes a light source for producing auniform sheet of light. The light source is oriented with respect to thetrack such that the workpieces occlude the uniform sheet of light uponpassing through the test section. The test section further has a videosystem for receiving the occluded uniform sheet of light, providingoutput signals related to the intensity of the occluded uniform sheet oflight incident on the video system, and a signal processor for receivingand processing the output signals.

U.S. Pat. No. 6,959,108 discloses an inspection system whereinworkpieces to be inspected are consecutively and automatically launchedto pass unsupported through the field of view of a plurality of cameras.As a workpiece passes through the field of view of the cameras, a sensoris activated which communicates with a computer system to activate thecameras to capture an unobstructed image, or image data, of theworkpiece. The image data is then analyzed by a computer program toverify whether the image data indicates that the workpiece does not meetestablished criteria and therefore is considered defective. If the imagedoes not meet the established criteria, the workpiece is rejected andsegregated from workpieces which have not been identified as defective.

U.S. Pat. No. 5,608,530 discloses a laser for producing a beam ofradiation which is then refined in cross-sectional dimension by use ofplano-cylindrical lenses. The refined beam of radiation falls incidenton a part to be measured. The unobstructed portion of the beam is thenbifurcated by a pair of reflective surfaces which produce non-parallelradiating beams. Each resulting beam comprises the unobstructed portionof radiation which has passed radially opposed halves of the part. Themagnitude of radiation present in each non-parallel radiating beam isthen measured.

U.S. Pat. No. 4,831,251 discloses an optical device for discriminatingthreaded workpiece by the handedness by their screw thread profiles. Thedevice present a pair of light beams which pass generally tangent to theworkpiece at angularly displaced positions. The light beams are inclinedto follow the helix direction of a given handedness of a workpiece. Uponaxial advancement of a workpiece through the device, a chopped outputfrom the photodetectors indicates that the handedness of the threadsmatches the inclination of the light beams. The oppositely threadedworkpiece, however, provides a generally constant DC output. Withappropriate signal processing electronics, an automatic system fordiscriminating workpieces by thread handedness is provided.

U.S. Pat. No. 5,383,021 discloses a non-contact inspection systemcapable of evaluating spatial form parameters of a workpiece to provideinspection of parts in production. The system causes parts to besequentially loaded onto an inclined track where they pass through atest section. The test section includes a length detection array formeasuring the length of the workpiece, which includes a sourcegenerating a sheet of light oriented in the longitudinal direction ofthe workpiece. The profile of the parts are evaluated by one or morelight sources also creating a sheet of light oriented transversed to thelongitudinal axis of the parts. Single channel photodetectors areprovided for each of the sources which provides an analog output of theextent to which each sheet of light is occluded by the part. Theseoutputs are analyzed through appropriate signal processing hardware andsoftware to generate length and profile data related to the workpiecegeometry.

U.S. Pat. No. 5,568,263 discloses a non-contact inspection systemcapable of evaluating spatial form parameters of a workpiece to provideinspection of parts in production. The system causes parts to besequentially loaded onto an incline track where they pass through a testsection. The test section includes a length detection array formeasuring the length of the workpiece, which includes a sourcegenerating a sheet of light oriented in the longitudinal direction ofthe workpiece. The profile of the parts are evaluated by one or morelight sources also creating a sheet of light oriented transverse to thelongitudinal axis of the parts. First and second pairs of single channelphotodetectors are provided for each of the light sources which providesa pair of analog outputs of the extent to which each sheet of light isoccluded by the part, as well as an ability to eliminate noise orscintillation caused by a point source of light, for example with alaser light source. These outputs are analyzed through appropriatesignal processing hardware and software to generate length and profiledata related to the workpiece geometry.

U.S. Pat. No. 4,852,983 discloses an optical system which simulates theoptical effect of traveling over a large distance on light travelingbetween reference surfaces.

U.S. Patent Application Publication No. 2005/0174567 discloses a systemto determine the presence of cracks in parts. The presence of cracks isdetermined through the use of an imaging device and illumination source.The part is moved along a track where it is sensed by a position sensorto initiate the inspection. The illumination source projects a sheet oflight onto the part to be inspected. The line formed by the intersectionof the sheet of light and the part is focused onto the imaging device.The imaging device creates a digital image which is analyzed todetermine if cracks are present on the part.

U.S. Patent Application Publication No. 2006/0236792 discloses aninspection station for a workpiece including a conveyor, a mechanism forrotating the workpiece, and a probe. The conveyor includes a fixture forlocating the workpiece and the conveyor is configured to translate theworkpiece in a linear manner. A mechanism, such as a belt, engages theworkpiece thereby rotating the workpiece within the fixture. The probeis configured to indicate if the workpiece conforms to quality criteria.To facilitate inspection while the conveyor translates the workpiece,the probe is attached to a stage where the stage is configured to movethe probe synchronously with the workpiece over an inspection region.

U.S. Pat. Nos. 5,168,458 and 5,170,306 disclose methods and systems forgaging threaded fasteners to obtain trilobular parameters.

Other U.S. patents related to the invention include: U.S. Pat. Nos.4,315,688; 4,598,998; 4,644,394; 4,852,983; 4,906,098 and 5,521,707.

SUMMARY OF THE INVENTION

An object of the present invention is to provide improved opticalmodules and method of assembling same.

In carrying out the above object and other objects of the presentinvention, a method of precisely assembling an optical module isprovided. The method includes the step of providing a set of opticalcomponents including a plurality of beam shaping components forconverting at least one initial shape in cross section of a laser beamto at least one desired shape in cross section and an optical mount forsupporting the set of optical components. The method further includesholding and locating the optical mount relative to a reference axis. Themethod still further includes holding the optical components in positionrelative to the optical mount during the step of holding and locatingwherein the positions of the beam shaping components are initialpositions. The method includes directing a laser beam having the atleast one initial shape at the set of optical components during the stepof holding the optical components. The method further includes adjustingthe initial positions of the beam shaping components to obtain finalpositions of the beam shaping components relative to the optical mountduring the step of directing until the laser beam has the at least onedesired shape. The method still further includes fixedly securing theoptical components to the optical mount wherein the optical mountsupports and is fixedly secured to the beam shaping components in theirrespective final positions after the step of adjusting when the laserbeam has the at least one desired shape.

The at least one initial shape may be circular or nearly circular incross section and the at least one desired shape may be generallyrectangular in cross section.

Divergence of the laser beam having the desired, generally rectangularshape may be less than or equal to a predetermined maximum value whichmay be 1.3 mrad or less.

The beam shaping components may include a first lens held by the opticalmount to move therewith during the step of holding the opticalcomponents and the method may further include the step of adjustinglinear position of the optical mount and the held first lens along thereference axis.

The method may further include measuring width of the laser beam at aplurality of distances from the first lens during the step of directingto obtain a plurality of width measurements. The laser beam has thedesired, generally rectangular shape when the width measurements aresubstantially equal.

The desired, generally rectangular shape may have a substantially flatprofile.

The beam shaping components may include a first lens and the step ofadjusting may include the step of adjusting angular position of thefirst lens relative to the optical mount until a profile of the laserbeam is substantially flat.

The step of adjusting may include the step of adjusting the horizontalposition of the first lens relative to the optical mount until powerdistribution in the laser beam is balanced within the profile.

The beam shaping components may include a second lens and the step ofadjusting may include the step of adjusting angular position of thesecond lens relative to the optical mount until a profile of the laserbeam is level.

The step of adjusting may include the step of adjusting horizontalposition of the second lens relative to the optical mount until thelaser beam is centered along the reference axis.

The beam shaping components may include a third lens and the step ofadjusting may include the step of adjusting vertical position of thethird lens relative to the optical mount so that the laser beam islocated in the same vertical position at a plurality of distances fromthe third lens.

The step of adjusting may include the step of adjusting angular positionof the third lens relative to the optical mount.

Further in carrying out the above object and other objects of thepresent invention, a light plane generator module assembled byperforming the steps of the method is provided.

The beam shaping components may include first, second and thirdcylindrical lenses.

Still further in carrying out the above object and other objects of thepresent invention, a light plane receiver module assembled by performingthe steps of the method is provided. The at least one initial shape is apair of spaced, generally rectangular shapes in cross-section and the atleast one desired shape is a pair of spaced circular shapes incross-section.

The set of optical components may include a pair of spaced lenses and acorresponding pair of spaced photodetectors.

The spaced lenses may include a pair of spaced spherical lenses.

The set of optical components may include a pair of spaced aperturedelements disposed between the pair of spaced spherical lenses and thepair of spaced photodetectors.

The above object and other objects, features, and advantages of thepresent invention are readily apparent from the following detaileddescription of the best mode for carrying out the invention when takenin connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic perspective view of a part inspection systemincluding measurement hardware;

FIG. 2 is a schematic, side elevational view of a part holder base (witha side plate of a container removed) and an upper tooling unit with apart held between the units and a calibration fixture or device mountedcolinear with the part;

FIG. 3 a is a perspective schematic view of the part holder base with adrive bit supported thereon and the calibration device suspendedtherefrom;

FIG. 3 b is a top plan view of the base of FIG. 3 a;

FIG. 3 c is a sectional view taken along lines 3 c-3 c of FIG. 3 b;

FIG. 4 a is a side elevational view of the calibration device or cone ofFIG. 3 a;

FIG. 4 b is a top plan view of the cone of FIG. 4 a;

FIG. 4 c is a sectional view taken along lines 4 c-4 c of FIG. 4 b;

FIG. 5 is a top plan view of an optical head with its top cover plateremoved in order to provide an interior view of the optical head;

FIG. 6 is a schematic perspective view of a safety enclosure which mayenclose the basic measurement hardware of FIG. 1;

FIG. 7 is a side elevational view, partially in cross-section, of theupper tooling unit of FIG. 2;

FIG. 8 is a schematic block diagram which illustrates basic beam linesubsystem components wherein a laser generates a laser beam, a mirrorreflects the laser light beam, and a light plane generator modulegenerates a laser light plane which is directed or projected onto apart;

FIG. 9 is a schematic perspective view of a cylindrical part which isintersected by a projected laser light plane;

FIG. 10 is a schematic view, partially broken away, of various positionmeasurement system components together with the optical head which ismounted on a stage to move therewith just prior to scanning thecalibration device or cone;

FIG. 11 are graphs of various raw sensor signals generated when thebeginning edge of the calibration cone is scanned;

FIG. 12 is a schematic block diagram of various laser transmitters andreceivers contained within the optical head;

FIG. 13 is a top plan schematic view of a part illuminated by multipleplanes of laser light with various tangential shadow rays and points;

FIG. 14 is a graph of sensor height data created with a threaded partsuch as a standard thread plug go gage;

FIG. 15 is a symbolic sketch of the calibration cone's profile withvarious usage regions;

FIG. 16 is a schematic block diagram which illustrates raw sensor dataprocessed with calibration data to obtain calibrated sensor data;

FIG. 17 is a thread model superimposed on a graph which illustratesrough thread peaks, troughs (or more correctly “crests” and “roots”) andcrossings obtained by the intersection of a pitch diameter line withflank lines;

FIG. 18 is a graph similar to the graph of FIG. 17 wherein selectedthread concepts are illustrated;

FIG. 19 is a single thread form which illustrates thread flank lines, athread flank line data extraction region and a wire position;

FIG. 20 is a schematic diagram which illustrates a 3-point distancemeasurement between a reference line defined by a pair of virtual wirecenters on one side of a threaded part and a single wire center on theopposite side of the part;

FIG. 21 is a screen shot of a graph of sensor height data with virtualwires in roots and with intermediate data illustrated;

FIG. 22 is a screen shot which illustrates an enlarged thread pitch withintermediate data;

FIG. 23 is a top plan schematic view of a cylindrical part with a beamof light deflected at an angle θ_(refl)=2θ by a perfectly reflectingsurface of the part;

FIG. 24 is a top plan schematic view of a part which scatters light froma plane of light which scattered light is blocked by light planereceiver aperture slits;

FIG. 25 illustrates a conical shape (i.e., frustum) utilized in thecalibration fixture;

FIG. 26 is a sketch, partially broken away, similar to the sketch ofFIG. 15 which shows a schematic outline of the sensor signal produced bythe calibration cone and various support structures;

FIG. 27 is a sketch which illustrates a full open signal level computedfrom data in the full open estimation region;

FIG. 28 is a graph which shows the data regions of FIG. 27;

FIG. 29 is a sketch which illustrates cone projection geometry;

FIG. 30 is a sketch of α₁, α₂, α₃, α₄ which are the projections of{right arrow over (α)} on the y′-axis for the θ=22.5, 67.5, 112.5, 157.5degree laser sensor systems; the positive y′-axis is the right sensorand the negative y′-axis is the left sensor direction;

FIG. 31 is a schematic perspective view of an adjustment fixture forassembling optical modules;

FIG. 32 is a graph illustrating a relatively non-level laser beamprofile;

FIG. 33 is a graph illustrating a relatively flat laser beam profile;

FIG. 34 is a graph illustrating a relatively collimated laser beamprofile;

FIG. 35 is a graph illustrating a distorted, generally rectangular laserbeam profile;

FIG. 36 is a graph illustrating a non-distorted, generally rectangularlaser beam profile;

FIG. 37 is an exploded perspective view of a typical transmitter moduleincluding its transmitter mount and its supported set of opticalcomponents;

FIG. 38 is a back view of the transmitter mount of FIG. 37;

FIG. 39 is a sectional view of the transmitter mount taken along lines39-39 of FIG. 38;

FIG. 40 is a top plan view of the transmitter mount;

FIG. 41 is a schematic perspective view of the assembled transmittermodule;

FIG. 42 is a front view of the assembled transmitter module;

FIG. 43 is a sectional view of the transmitter module taken along lines43-43 of FIG. 42;

FIG. 44 is a top plan view of the assembled transmitter module;

FIG. 45 is an exploded perspective view of a laser beam steering mirrorassembly;

FIG. 46 a-46 d are top plan views of the assembly of FIG. 45 and havingdifferent angular portions for use in the optical head of FIG. 5; and

FIG. 47 is an exploded perspective view of a typical receiver moduleincluding its receiver mount and its supported set of opticalcomponents.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The overall system described herein is often referred to as “Laser Lab.”Laser Lab is a trademark of the assignee of this application. It is tobe understood that numerous inventions are described herein, only someof which are claimed herein. The other disclosed inventions are claimedin the applications noted in the Cross Reference to Related Applicationspart of this application. It is also to be understood that a number ofwords and phrases are explained in a Glossary portion of thisapplication. The Glossary explains but does not unduly limit the wordsand phrases contained therein.

Laser Lab—Physical Overview

The Laser Lab system (i.e., FIG. 1) includes several physical subsystemsor units.

A PC tower unit (i.e., FIG. 8) contains a computer and a number ofadditional control electronics modules. It has a rear panel withconnectors for light curtain/safety electronics, motor control, linearencoder, measurement signals, and optical head control. The PC towerunit hosts the application program which presents a user interface to anoperator of the Laser Lab system.

Part holder and upper tooling units (FIGS. 1, 2 and 7) secure or receiveand retain a part in place for measurement. The upper tooling unitincludes a stainless steel rod with a spring loaded tip that can move upand down to accommodate a wide variety of part sizes. The part holderunit has a base to support the part or unit under test (UUT) and acalibration cone or device (i.e., FIGS. 3 a-4 c). The calibration coneis used to measure the relationship between a light sensor output andthe physical measurements represented by the outline dimensions of thecone. The calibration cone or device is not strictly speaking a cone butrather includes a number of frustums (i.e., portions of cones) andcylinders.

An optical head (i.e., FIGS. 1, 5 and 10) is a sealed system containinga number of components including optical measurement components (i.e.,FIGS. 5 and 12). A set of (4) laser beam lines (one of which is shown inFIG. 8) generate and measure 4 planes of light.

A slide/base unit (i.e., FIGS. 1 and 10) moves the optical headvertically up and down to make part measurements. On every scan theoptical head's (8) sensors measure shadow images of both the calibrationcone and of the part (UUT) (i.e., FIGS. 8, 9 and 13). Each complete scanthus contains both calibration data and measurement data, yielding asystem that is especially immune to time variations in measurementconditions.

Referring again to the drawing figures, FIG. 1 is a schematicperspective view of the Laser Lab system, generally indicated at 10,including the basic measurement hardware of the system 10. Shown are theoptical head, generally indicated at 12, the part holder/upper toolingunits, generally indicated at 14 and 16, respectively, and thebase/slide unit, generally indicated at 18, which, as also shown in FIG.10, includes a motor 20 coupled to a lead screw 22 which, in turn, iscoupled to a saddle 24 slidably supported by a bearing. The saddle 24 iscoupled to the optical head 12 to move the optical head 12 linearlyalong a vertical stage axis 28 (i.e., FIG. 9). Movement of the stage issensed by a linear encoder 30 which will be described in greater detailhereinbelow.

FIG. 2 is a schematic, side elevational view of the part holder andupper tooling units 14 and 16, respectively. The upper tooling unit 16includes a rod 32 which is manually movable along a central axis of therod 32 in up and down directions by an operator of the system 10. Theupper tooling unit 16 also includes a spring-loaded part clamp 34 havinga tip 35 which retains a part 36 to be inspected at one of its endsurfaces 38. Also illustrated in FIG. 2 are a calibration cone ordevice, generally indicated at 40, and a base or top plate 42 of thepart holder/base unit 14.

FIG. 3 a is a perspective schematic view of the part holder base unit14. FIGS. 3 a-3 c show the calibration cone 40 and a part holderassembly including a support 44, a collar 46, a bit holder 47 into whicha Torx bit 48 (i.e., Tx40) is inserted for engagement with a recessedportion of the end surface. A thread plug gage (instead of a recess bitfor a bolt) may be provided. The end user screws on a nut all the way tothe end of the plug gage. This ensures the threads of the nut are good.Then the system 10 measures the rest of the outer diametercharacteristics of the nut.

A cap 49 covers the assembly. FIGS. 3 a-3 c also show various mechanicalpieces or parts including a mounting plate 50, a riser 52, a bottomplate 54, the top plate 42, an adapter plate 58 for suspending the cone40 at the lower surface of the top plate 42 and a pair of posts 56 forsupporting the top plate 42 at a spaced position above the bottom plate54.

FIG. 4 a is a side elevational view of the calibration cone 40. Asdescribed in detail below, the calibration cone 40 has a preciselymanufactured shape that is utilized in measuring the relationshipbetween raw digitized sensor signals and calibrated physical dimensions.Typically, the cone 40 is sent to a certified laboratory which inspectsthe cone 40. The laboratory then provides a long form certification thatis traceable to NIST.

FIG. 4 b is a top plan view of the cone 40 of FIG. 4 a and FIG. 4 c is aside sectional view of the cone 40 taken along lines 4 c-4 c of FIG. 4b.

FIG. 5 is a top plan view of the optical head 12 with its top coverplate (60 in FIG. 1) removed in order to provide an interior view of theoptical head 12. The head 12 is attached to the saddle 24 to movetherewith. Shown adjustably mounted on the bottom base plate 61 of thehead 12 are laser steering mirrors 62, 4 lasers 64 for generating laserbeams, preferably having a wavelength of 650 nm, light plane generatoror transmitter modules, generally indicated at 66, for converting thegenerated laser beams into corresponding light planes, and light planereceiver modules, generally indicated at 68.

The calibration cone has a central hole, preferably about 1/16″, that isutilized extensively during assembly of the cone/part holdersub-assembly to the slide base unit as illustrated in FIG. 4 c.

At that time, a thin rod (“calibration fine centering rod”) is insertedthrough the cone's central hole when the optical head is in the “down”position. The rod defines the center of the calibration cone moreprecisely than the constant diameter region-O which has a ⅛″ diameter.When the head is in the “down” position the light planes pass below thecone, nominally unblocked.

To use the rod, the part holder (44 in FIG. 3 c) is removed from thecone/part holder assembly. Then the rod can be inserted from the top,through the plate 42, and through the cone 40, until it touches thebottom plate 54. The rod is long enough that when it is fully inserted,touching bottom plate 54, a length extends above plate 42, sufficient tomanipulate the rod by hand, moving it up or down.

Moving the “calibration fine centering rod” into and out of the lightplanes allow one to determine if the light plane center line passesthrough the rod. One observes the laser sensor outputs as the rod movesinto and out of the beam. When both left and right sensor outputs show aslight reduction from the effect of the rod blocking the light beam,then the light beam center line passes through the centering rod.

The calibration cone is manufactured with special instructions tofabricate the central hole so that there is a “slip fit” of the“calibration fine centering rod” in the calibration cone's central hole.One rod is paired with each calibration cone, by the cone fabricator.

The use of the “calibration fine centering rod” makes it possible tomeasure whether or not the cone/part holder center is aligned with theoptical head's 4-beam intersection point.

To move the cone/part holder with high precision and complete thealignment/centering operation, four fine-pitch “pusher screws” areutilized. Each “pusher screw” is mounted to the triangular base of theslide/base unit (FIG. 1). There is one “pusher screw” for each of thefour directions of movement of the cone/part holder plate 50 on thetriangular base, North, East, South, and West.

After the cone/part holder assembly is centered to the optical head beamcenter, the four “pusher screws” are tightened, maintaining the baseplate 50 position via compression. Then the “pusher screw” lockingsleeves are tightened. Then the base plate 50 hold down screws aretightened. The “pusher screw” locking sleeves can also be secured withan appropriate glue. The above-noted alignment process centers thecalibration cone precisely, with respect to the beam center of theoptical head.

FIG. 6 is a schematic perspective view of a safety enclosure, generallyindicated at 70, which encloses the Laser Lab basic measurement hardwareof FIG. 1 which sits inside the enclosure 70. A light curtain isgenerated by a light curtain transmitter 72 and is received by a lightcurrent receiver 74 to guard a physical access opening 76 to the LaserLab. The enclosure 70 and its hardware monitor circuitry (not shown)guarantee that the moving optical head 12 will be stopped before a usercan physically make contact with it.

FIG. 7 is a side elevational view, partially in cross-section, of theupper tooling unit 16. The long rod 32 of the tooling unit 16 can bemoved up/down by almost 9″ (for example) to accommodate a wide range ofUUT sizes when an operator manually grips a rod handle 78 mounted at aproximal end 80 of the rod 32. The operator can release an upper or alower friction release clamp 82 and 84, respectively, of the toolingunit 16. Support brackets 86 of a support structure, generally indicatedat 88, hold the upper tooling unit 16 to the base/slide unit 18. Rodguides 90 supported by a guide support 92 which, in turn, is supportedby the support brackets 86 and plates 94, hold the rod 32 precisely andalso allow it to slide smoothly up/down. Raising the top release clamp82 allows the rod 32 to move down, while lowering the bottom releaseclamp 84 allows the rod 32 to move up. The spring loaded part clamp 34has a threaded hole 96 for threadably receiving and retaining one of anumber of possible tips such as the tip 35 that contacts the part 36 tobe inspected.

Laser Lab—Basic Measurements Overview

The Laser Lab is a system for measuring the dimensions of a variety ofmanufactured parts and/or assembled parts such as parts manufactured inthe fastener industry. These parts are typically formed from cylindricalstock by roll and impact die forming methods or by cutting with lathes.The final part can have forms that are built up from basic shape units,such as circular or tapered cylinder, threaded cylinder, or additionalsimple shapes such as Trilobe cylinder or hex cylinder.

Substantially all of the measurements obtained with the Laser Lab systemare based on two basic components: (1) the height of a surface from thelight plane split line, and (2) the positions along the optical head'sstage axis corresponding to the various heights.

In addition, the Laser Lab system performs multiple measurements from(4) different measurement directions. However, it is to be understoodthat, depending on the part, the measurements can be taken from as fewas two measurement directions. In some cases, as many as fivemeasurement directions may be needed. This capability of being able toobtain multiple measurements from multiple directions allows the LaserLab to explicitly utilize 3-D shape information, especially in themeasurement of threaded cylinders and non-cylindrical shapes.

Surface Height Relative to the Measurement Axis

In this section the measurements made by a single laser are describedwith reference to FIG. 8. The subsystem that makes these measurements iscalled a beam line. (4) beam lines are contained in one optical head asshown in FIG. 5.

The light plane generator module 66 of FIG. 8 creates a light plane froma single laser beam generated by the laser 64. The part creates shadowedand unshadowed regions in the light plane. The light plane receivermodule's left and right receiver (Rcvr) components (i.e., opticaldetectors) convert the amount of light on the left and right sides ofthe laser split line into separate left and right electrical signals,respectively, which are digitized by receiver electronics when thereceiver electronics receives a trigger or sampling signal from encoderelectronics as described herein.

The PC analyzes the digitized signals from the receiver electronics andcomputes left and right sensor heights. The measurement computationutilizes a sensor height calibration (described later herein) to convertthe raw digitized sensor signals into calibrated heights, measured inmm.

The surface height measurement is further illustrated in FIG. 9. Theleft and right sensor heights are the amount of light blockage in thelight plane, on both left and right sides. The light plane is nominally,but not exactly, perpendicular to both the stage axis 28 and the partaxis.

Naming of Signals

In what follows the measurements that are based on light detections inthe left or right receivers are referred to as left or right sensorsignals or measurements, depending on the context. With (4) laser beamlines there are (8) sensor signals in the system 10. When referring tothese signals in the entire system 10 the laser number is added tospecify the beam line. Thus names for the sensor signals range fromlaser-1, left sensor through laser-4, right sensor.

As previously mentioned, FIG. 8 is a block diagram which illustrates thebasic beam line system components including the laser 64 which generatesa laser beam, the mirror 62 which reflects the laser beam and the lightplane generator module 66 which generates a light plane which isprojected at the part (UUT). The part blocks a portion of the lightplane. The components also include the light plane receiver module 68having left and right receivers or photo detectors. However, it is to beunderstood that instead of two detectors, a line scan camera, an LCDcamera or other optical detector device may be provided.

FIG. 9 which is a schematic perspective view of a cylindrical part whichis intersected by a projected laser light plane illustrates light planemeasurement geometry.

Stage Position of the Optical Head

The optical head 12 generates (4) spaced apart laser light planes and istranslated up and down by the moving stage system illustrated in FIG.10. The light planes are projected perpendicular to the Z-axis definedby the movement of the stage (i.e., the stage axis 28). The linearencoder 30 preferably produces one measurement trigger signal pulseevery 4 μm of stage movement. The signal is sent to the receiverelectronics module. At the receiver electronics module, each measurementtrigger signal pulse causes each sensor signal (8) to be digitized andstored into the PC's memory.

This combination of the linear encoder electronics, the measurementtrigger signal, and the receiver electronics creates a sequential recordof sensor digitized raw signals when the stage moves from bottom to top.The resulting record can be interpreted as a record of light blockage byeither a part or the calibration device at a known series of positions,spaced 4 μm apart. In a preferred system the total linear length ofstage movement is about 235 mm.

The position measurement system described above measures positionintervals, but generally not repeatable positions. The moving stage isstopped near the top and bottom travel limits by electronic limitswitches (not shown). Reaching the top or bottom travel limit switch bythe stage causes the motor 20 to stop. However the actual stoppedposition is only approximate, since the limit switches are not precisioninstruments calibrated to the encoder 30 and since the distance thestage requires for stopping depends on the speed of travel. In practicethis results in an uncertainty in the stopping position that can be aslarge as 500 μm.

To make a predictable starting position a light blockage signal isanalyzed to extract the index position of the beginning of thecalibration cone 40 in each sensor's digitized raw signal. The beginningof the cone 40 is preferably formed by a 0.125″ diameter cylinder 96 ofthe cone 40 about 0.2375″ long. The raw sensor signal is at a high levelas each light plane moves towards the beginning edge of the cone 40,followed by a sharp step decrease in the response, and finally followedby a constant response along the length of the cylinder 96. The analysissoftware locates the midpoint of the sharp step in the response and usethat index position to set the position zero for the sensor stage axisposition. An example of the raw sensor signal is shown in FIG. 11.

It is found in practice that this technique can reduce the positionaluncertainty of fixed positions on the part or on the cone 40 to anamount (5 μm) that is much less than the uncertainty in the stoppingposition (˜300 μm).

FIG. 10 is a schematic view, partially broken away, of various positionmeasurement system components together with the optical head 12 which ismounted on the stage to move therewith to scan the calibration cone 40(and later the part). As previously mentioned, the motor 20 is coupledto the lead screw 22 which drives the moving stage and, consequently,the optical head 12 in a direction along the stage axis 28 dependent onthe direction of rotary motion of the lead screw 22. The linear encodersenses the linear position of the moving stage along the stage axis 28and provides a corresponding output signal to the linear encoderelectronics. The electronics generate a trigger signal about every 4 μmof stage movement along the stage axis. The trigger signal is receivedand processed by the receiver electronics as previously described.

FIG. 11 illustrates graphs of various raw sensor signals generated atthe beginning of the calibration cone 40 (i.e., the cylinder 96). Shownare the signals from sensors laser-1, left, laser-1, right, laser-4,left, and laser-4, right. The raw sensor signals are plotted with thehighest values at the bottom of the graphs with values decreasingupwards. The left most step in the laser-1, left response represents ajump of about 2650 digitization units in the raw sensor response.

The plane of laser light from laser-4 is blocked by the calibration cone40 before the plane of laser light from laser-1 on this upward movingscan. That is because the light plane for the laser-1 is the lowest inthe optical head 12 as shown in FIG. 10.

Multiple Beam Optical Head

As previously mentioned, the optical head 12 contains (4) beam linesubsystems. The subsystems are aligned on a common central axis. Lookingdirectly down on the optical head 12 the beam line light plane splitlines preferably intersect at a common point as shown in FIG. 12. Theangles of the beam lines, relative to the front of the optical head baseplate, are 22.5, 67.5, 112.5, and 157.5 degrees.

This arrangement, combined with the mechanical scanning of the lightplanes, results in (8) outline images of the part, one per sensor. FIG.13 shows the geometry of a typical situation when light from the (4)beam lines (light planes) intersects a part with a circular crosssection. For each beam line, two shadow rays graze (are tangent to) thesurface of the part, marking the left and right limits of the part'sshadow. The points of intersection between the shadow rays and thepart's surface are called shadow points.

The calibrated distance between the shadow points and the light planesplit line is the sensor height as shown in FIG. 9. FIG. 14 shows thesensor height plotted for both the left and right sensors of laser-1 fora full scan of a threaded part. This plot is essentially an orthographicprojection of the part, in a viewing direction aligned with the laser-1,laser split line vector.

As previously mentioned, FIG. 12 is a top plan schematic view of themodules 66 and 68 of the optical head 12 with its top plate 60 removed.The laser split line for each transmitter module 66 is indicated as adashed ray which has an arrow head which illustrates the direction oftravel of the light beam and the plane of laser light.

FIG. 13 is a top plan view of a cylindrical part which is scanned by 4planes of laser light including shadow rays which are tangent to thepart at shadow points.

FIG. 14 is a graph of sensor height data created from a threaded partsuch as a standard thread plug go gage. Shown is laser-1 data; the leftsensor data is plotted on the top half of the image, the right sensordata is plotted on the bottom half of the image.

Calibration Cone

The calibration cone 40 is a device which has a precisely manufacturedshape or outer surface which is scanned to obtain calibration datawhich, in turn, is used to convert sensor raw digitized signals tocalibrated sensor height measurements. The cone 40 is a rotationallysymmetric with several distinct regions, each designed to perform adifferent calibration function.

A symbolic sketch of the calibration cone's form is shown in FIG. 15.The calibration cone is designed with “usage regions”, or specificshapes designed to accomplish specific calibration goals. These regionsare listed in Table 1 below. Each “usage region” is designed to allow aspecific piece of calibration information to be extracted from thescanned data set. The following sections briefly describe these piecesof calibration information.

Stage Position Alignment of Different Sensors

Each of the lasers 64 is mounted in the optical head 12 at a differentheight offset to the base plate 61 of the optical head 12 as describedwith reference to FIG. 10. The height offsets are not as precise asdesired and depend on detailed optical and mechanical adjustments of theoptical head 12 and its optical modules 66 and 68. The height differencebetween adjacent channels might be as large as 500 μm. The only way tomake sure that the calibrated sensor stage positions refer to the samephysical objects at the same stage positions is to measure a commonphysical position.

The precise location of the middle of the “begin cone edge” marks thecommon zero (0) of each laser sensor's calibrated stage position.

Sensor Height Zero Position Alignment of Different Sensors

It is important to select a common zero position in a plane that isperpendicular to the stage axis 28 and aligned parallel to the lightplanes in the optical head 12. The position zero that is selected in thecalibration process is the intersection of the light planes with thecenter of the small cylinder 96 at the beginning of the calibration cone40 (i.e., FIG. 10).

A light plane split line defines a natural zero for the sensor heightmeasurement, but the (4) light plane split lines do not necessarilyintersect in a single point as illustrated in FIG. 12. The optical headalignment process only ensures that the light plane split linesintersect within a 1/16″ (1587.5 μm) cylinder.

Adding the position offset described above makes the lines defining theposition zero of each sensor's calibrated height intersect at the centerof the small cylinder 96 at the beginning of the calibration cone 40.Measuring the center, after calibration, typically gives a centrallocation that is less than 1 μm from (0,0).

Calibration Cone Aspect Vector, Relative to Stage Axis

The calibration cone's central axis is not necessarily exactly alignedwith the axis 28 of the stage motion, due to tolerance stackup on a longpath from cone 40, to part holder, to base plate, to slide support,etc., and finally to the slide.

By measuring the sensor heights of the center of two calibration coneregions, “const diam-1” and “const diam-2”, the inclination of thecalibration cone's aspect vector relative to the stage axis 28 can bedetermined.

Typical measured angles for the calibration cone aspect vector relativeto the stage axis 28 are in the range (0 . . . 1 degree) or (0 . . .17.5 mrad).

Light Plane Angle, Relative to Calibration Cone

The laser light planes are not exactly perpendicular to the calibrationcone aspect vector. In order to know the angle between the light planesand the calibration cone 40, the “multi-step” region is analyzed. Signalprocessing software can very accurately measure the position of each oneof the set of 5 step edges in the “multi-step” region. The distancebetween the 5 step edges is precisely known. With this information theangle of the light plane relative to the cone aspect vector can becomputed. This angle is important in determining exactly how the lightplane intersects the calibration cone 40 and thus in extractingcalibration information from the data.

Typical measured angles of the light plane relative to the calibrationcone 40 are in the range (0 . . . 0.75 degree) or (0 . . . 13.1 mrad).

Sensor Height Calibration

The output of the laser scanned measurement is a record of the sensordigitized raw signals for each sensor. To make sensor heightmeasurements in physical coordinates the raw signals need to beconverted to sensor heights.

Two regions on the calibration cone, “const slope-1” and “const slope-2”provide this information. For example, the diameter of the intersectionbetween a laser light plane and “const slope-1” region varies between0.125″ and 0.750″. The exact diameter can be computed by knowing thedistance between the laser light plane and the beginning of the “constslope-1” region, since the region is manufactured to high precision.

Based on the diameter of intersection and the laser sensor outputs,calibration tables may be constructed to convert digitized raw signalsto calibrated sensor heights.

Measurement of the Sensor “No Blockage” Signal Level

The raw signal level with no cone 40 or UUT in the light plane sensorbeam is also measured. For small parts it is often required toextrapolate the sensor raw signal to sensor height conversion table tosmaller heights than are measured on the cone. The extrapolation iscarried out with less accuracy than more direct measurements, but theextrapolation is very useful, especially for parts that are onlyslightly smaller than the begin cone cylinder 96 (“const diam-0” region)of the cone 40 or parts that are offset from the calibration conecentral axis.

The “no blockage” signal level is also required in order to correctlyfind the beginning of the cone 40.

Finally, excessive variability of the “no blockage” signal level is asignature of variability of the light output of the laser 64 in a beamline. This variability of the “no blockage” signal level is monitored togenerate a signal which indicates that the apparatus which generates thebeam line requires repair or is temporarily unable to carry out highprecision diameter measurements.

This measurement goal does not involve measurement of the calibrationcone 40. However, it is required to interpret calibration cone analysis.It is made possible by the physical design of the part holder base asillustrated in FIG. 3 a, and by the alignment of the bottom position ofthe optical head 12 on the base/slide to the part holder base.

TABLE 1 Calibration Cone - Usage Regions (i.e., FIGS. 15 and 26) RegionDescription begin cone edge Begin cone step edge. Measurement of theprecise step edge position makes it possible to align measurements madewith different laser sensors. The precise edge location is each sensor'sstage axis zero (0) coordinate. const diam-0 Constant diameter region-0,diameter 0.125″. The central axis of this region, projected to thesensor is the common physical sensor height zero (0) coordinate. constdiam-1 Constant diameter region-1, diameter 0.750″. The central axis ofthis region is utilized for estimates of the 3-D cone aspect vector,relative to the stage axis 28. const diam-2 Constant diameter region-2,diameter 0.750″. The central axis of this region is utilized forestimates of the 3-D cone aspect vector, relative to the stage axis 28.const max diam Constant maximum diameter region, diameter 1.500″. Marksthe boundary between the constant slope region and the multiple stepregion. const slope-1 Constant slope region-1. Measurements in thisregion establish the raw sensor output corresponding to diameters in theinterval 0.125″ and 0.750″. const slope-2 Constant slope region-2.Measurements in this region establish the raw sensor outputcorresponding to diameters in the interval 0.750″ to 1.500″. multi stepMultiple step region. Measurements of the precise step edge positions ofthe (5) equal height edges allow a rough estimate to be made of theangle of the Laser Light Plane relative to the cone aspect vector. slopeedge-0,1 Slope edges 1 and 2. The edge positions mark the boundaries ofthe const slope-1 region. slope edge-2,3 Slope edges 2 and 3. The edgepositions mark the boundaries of the const slope-2 region.

Other embodiments of the calibration cone 40 are possible while stillmeeting the general measurement goals of the Laser Lab system 10.

Calibration Measurement Goals

These are goals that calibration analysis of the cone data meet:

-   -   1) stage position alignment of different sensors    -   2) sensor height zero position alignment of different sensors    -   3) calibration cone aspect vector, relative to stage axis 28    -   4) light plane angle, relative to the calibration cone 40    -   5) sensor height calibration    -   6) measurement of the “no blockage” signal level.

In another embodiment of the calibration cone 40, changes to overallminimum and maximum width of the cone 40 can be made.

The calibration cone's “point design” allows one to extract calibrationdata relevant to measuring parts in the diameter range 0.125″ to 1.500″.

The system can be designed for smaller or larger parts wherein widthmeasurement limits could be changed. A small compact system formeasuring a range of smaller diameters could utilize a calibration cone40 with minimum and maximum diameters of 0.065″ and 0.500″ for example.

In another embodiment, changes to overall length of the cone 40 can bemade.

The calibration cone's “point design” is specified as a compromisebetween ease of analysis and physical compactness. The lower the slopein the “const slope” regions, the more precise the data that isextracted. This is due to two reasons. First, dividing up a slopingregion into “bins” and then determining the raw data to heightconversion factor within an individual bin is more accurate when themechanical diameter varies least within the bin.

Second, inaccuracies in determining the light plane twist angle or lightplanes that are not flat are multiplied into inaccuracies in the rawdata to height conversion factors by the mechanical slope of the constslope regions in the calibration cone 40.

In yet another embodiment, changes to slope in “const slope” regions canbe made. As noted in the above-noted discussion, lower slopes in the“const slope” regions translates to more accuracy in the sensor rawdata-to-height conversion factor tables.

A “point design” directed towards the goal of smaller parts and towardsa more compact system design might have a cone 40 with the same length,but 3× smaller width dimensions. This would allow more precision inmeasuring smaller parts with smaller width light planes.

In another embodiment, the cone 40 can be supported either “point down”or “point up”. The mounting direction does not matter, the calibrationmeasurement goals can be met with either orientation. However, themounting method should still allow a region of “no blockage” sensorsignal to be measured during each up/down scan.

In yet another embodiment, different number of steps in “multi-step”region can be provided. The number of steps in the “multi-step” regioncan be varied and the dimensions of the steps can be changed.

The calibration analysis that determines the light plane twist angleuses the difference in position between steps detected in the left andright sensors. Having more steps makes the determination more precise.

In another embodiment, changes to the number, position, or diameter ofthe “constant diameter” regions can be made. The calibration cone aspectvector is measured by analysis of the central axis of two constantdiameter regions, “const diam-1” and “const diam-2”. Each region is thesame diameter. Determination of the location of their 3-D center allowdetermination of the central axis of the calibration cone 40, from thetwo 3-D center points. It is important that both regions be the samediameter, to minimize the effect of diameter measurement errors on thecalibration cone axis vector.

The regions could be a different diameter, either smaller or larger.There could also be more than two regions. Then a line could be fitthrough 3 or more 3-D points to determine the calibration cone axisvector. It is important that there be at least two regions, one regiondoes not typically determine the calibration cone axis vector.

Laser Lab Calibration Analysis Overview

What is now described is a process oriented overview of the Laser Labcalibration procedure. In this procedure raw sensor data and geometricdescriptions of the calibration cone 40 are utilized to producecalibration data. This data set can be used later to produce calibratedsensor data from raw sensor data, as described in the “data processing”section, below.

Calibration Goals

As previously mentioned, the following is a partial list of themeasurement process goals that are met by the design of the mechanicalcalibration cone 40 and also by the calibration data analysis procedure:

-   G-1: stage position alignment of different sensors-   G-2: sensor height zero position alignment of different sensors-   G-3: calibration cone aspect vector, relative to stage axis 28-   G-4: light plane angle, relative to calibration cone 40-   G-5: sensor height calibration    Data Processing

After a laser lab scan of a UUT, (8) sensor digitized raw signals arestored, one each for the left and right sensors, repeated for each of(4) light planes. Each of these digitized raw sensor signals is a single“vector” or indexed list. The sensor raw signal vector has an “index”for every sample stored on the laser lab scan of the UUT; index 1999refers to the 1999-th sample taken during the scan. The value in thevector at index 1999 is the value of the 1999-th raw signal sample.

Production of Calibration Data

After the raw sensor signal vectors are stored in memory of the computeror PC, the vectors are analyzed to extract calibration information. Thisinformation is extracted from the part of the digitized raw sensorsignals that contains the image of the calibration cone 40. Theextracted information results in a number of tables and parameters,collectively called “calibration data”.

Production of Calibrated Sensor Data

Once calibration data has been successfully calculated it is utilized toproduce a new set of vectors called “calibrated sensor data”. The newcalibrated sensor data vectors contain two pieces of information at each“index”: the pair (calibrated stage position, calibrated height).

Calibrated Stage Position

Stage position for the N-th index in the calibrated sensor data vectoris the distance (in mm.) from the beginning of the calibration cone 40to the position where the raw sensor signal at the N-th index was taken.

First the raw sensor stage indices (the index of the raw sensor signalvector) are multiplied by the stage linear encoder spacing (4 μm in thecurrent system), producing raw sensor stage positions. Then the rawstage positions are referenced to the position of the beginning of thecalibration cone 40.

Finally the raw stage positions are corrected for laser tilt. The tiltcorrection depends on the height of the sensor data point. If the laserplane is slightly tilted, then any non-zero sensor height alsorepresents a slight change of stage position since the light plane,stage axis coordinate system is not orthogonal. After the correction thecalibrated sensor height, calibrated sensor stage position coordinatesystem is orthogonal. Having an orthogonal coordinate system makes latermeasurement analysis much simpler.

Calibrated Sensor Height

Calibrated sensor height for the N-th index in the calibrated sensordata vector is the distance (in mm.) from the center of the calibrationcone's beginning cylinder 96 to the shadow ray that produced the rawsensor signal at the N-th index.

As discussed herein above, the observations in the non-orthogonal stageposition, light plane coordinate system are corrected for the effects oflaser tilt.

Decimation to a Uniform Sampling Interval

The corrections for laser tilt result in a vector of calibrated sensordata where the stage position distances between adjacent index positionsin the vector can vary around an average value of 4 μm.

Since uniformly sampled data is much easier to work with for measurementanalysis, the calibrated sensor data vector is decimated or sampled touniformly sampled calibrated sensor data vector for measurementprocessing. In the current system the data is decimated to the originallinear encoder sample spacing of 4 μm.

FIG. 16 is a schematic block diagram which illustrates sensor dataprocessing. The figure shows the processing of raw sensor data withcalibration data to obtain uniformly sampled calibrated sensor data.

Calibration Analysis

Calibration analysis refers to the analysis of the raw sensor datavectors containing an image of the calibration cone 40. The output ofthe analysis is a set of tables and parameters called “calibrationdata”.

Rough Edge Processing

Rough edge processing discovers the presence and rough parameterizationof the signal edges in the raw sensor data.

Rough edge processing attempts to find the “pattern” of edges thatidentify the calibration cone 40 in the raw sensor data vector. Thispattern is schematically illustrated in FIG. 15.

Two types of edges are found. The first type of edge, a “step edge”,represents a vertical segment on the calibration cone 40. A step edgedetector finds one edge corresponding to the begin cone edge and (5)edges corresponding to the locations of the vertical segments in thecalibration cone's multi-step region.

A second type of edge, a “slope edge” represents the location where twostraight segments join with each segment having a different inclinationto the vertical. A slope edge detector looks for slope edges only inlocations where there is not a step edge. All step edges are also slopeedges. The slope edge detector finds a first unique slope edge at thelocation where “const diam-0” region meets “const slope-1” region, andin (3) other places.

Rough Edge Processing—Outputs

-   -   rough locations for step edges.    -   rough locations for slope edges.    -   confirmation that the calibration cone edge pattern is present        in the data.

If the rough edge processing step does not find the calibration coneedge pattern, then the calibration analysis process is stopped.

Precise Edge Processing

Precise edge processing finds the exact locations of step edges in thecalibration cone edge pattern. Precise edge processing utilizes outputsfrom rough edge processing to determine initial estimates for the edgelocations, which it then refines.

A detailed description of precise edge processing is located in AppendixB.

Precise Edge Processing—Outputs

-   -   precise location for begin cone step edge.    -   precise locations for (5) edges in “multi step” region.

The knowledge of the set of (8) begin cone step edges, one for eachsensor, completes calibration goal G-1: stage position alignment ofdifferent sensors. This data is stored in the calibration data andutilized to convert raw sensor data to calibrated sensor data.

Data Binning

At 4 μm per sampled point, there can be too much data to be effectivelyanalyzed for certain calibration processes. Data binning is the processof dividing up a set of sampled points, grouping the set into a smallerset of “bins”, each bin containing a number of adjacent sampled points.

For the tables relating the raw sensor data to calibrated sensor heightsbinning is utilized. For example, the “const diam-1” region on thecalibration cone 40 is about 12 mm long, ranging from 3.810 mm to 15.558mm along the cone axis. This would be about 3000 data points withoutbinning. At the nominal bin size of 0.2 mm this works out to about 60bins.

Another advantage of binning is that the data within the bin can beaveraged and checked for consistency.

Finally, the data bins are not constructed within a “guard” regionwithin 0.2 mm. of a detected edge. For the “const diam-1” region theedges “slope edge-1” and “slope edge-2” mark the boundaries of theregion and the “guard” region assures us that the boundary data bincontains only data from the uniformly sloping region.

Data Binning—Outputs

Four sets of data bins are produced, for each of (8) sensors:

-   -   two sets of data bins for the regions “const slope-1” and “const        slope-2”.    -   two sets of data bins for the regions “const diam-1” and “const        diam-2”.

The “const slope-n” data is used in the construction of the sensorheight calibration table. The “const diam-n” data is used as input datato the process that finds the position of the calibration cone's 0.750″diameter cylinder, for the cone aspect angle estimation process.

Laser Roll

Laser roll processing finds the angle between a light plane and thecalibration cone 40 in each laser's calibrated sensor coordinate system.

For each laser the precise edge locations for the (5) edges in the“multi step” region are obtained, one set for the left sensor andanother set for the right sensor. The difference between the left sensorand the right sensor edge positions can be used as input to a leastsquares estimate of the laser roll angle.

The detailed method of estimation is described in Appendix B.

The estimate of the laser roll angle assumes that the light plane isflat.

Laser Roll Processing—Outputs

-   -   (4) laser roll angles, relative to calibration cone 40,        projected to each laser's coordinate system.

The estimate of (4) laser roll angles completes calibration goal G-4:light plane angle, relative to calibration cone 40. These angles arestored into the calibration data.

Sensor Blockage and Tilt

The primary goal of the sensor blockage and tilt process is to generatea calibration table that relates the sensor raw signal values to thecalibrated sensor heights.

Achieving the primary goal is made difficult because the calibrationcone 40 may be mounted at an angle that may not be parallel to the stageaxis 28. If the cone angle is not parallel to the stage axis 28 then theinterpretation of exactly where the light plane hits the calibrationcone 40 depends on the angle between the calibration cone 40 and thestage axis 28.

To solve this problem an iterative process was created.

First, the sensor calibration tables were created, assuming the coneangle and stage axes 28 are parallel. Then using the newly createdsensor calibration table an estimate of the cone angle was made. Theprocess is repeated (4) times. The iterative process has been found toconverge in all cases. It is recommended that the mechanical alignmentof the cone aspect angle to the stage axis 28 be less than (1 degree).

This process is documented in more detail in Appendix B.

Sensor Blockage and Tilt—Outputs

The sensor blockage and tilt calibration process has two outputs.

-   -   (8) sensor raw data to calibrated sensor height tables, stored        in a calibration object.    -   a 3-D estimate of the angle between the calibration cone aspect        angle and the stage axis 28.

The (8) calibrated sensor height tables complete calibration goal G-5:sensor height calibration. The 3-D cone aspect angle meets calibrationgoal G-3: calibration cone aspect vector, relative to stage axis 28.

Sensor Height Table Extrapolation

The tables that correlates raw sensor data to calibrated sensor heightsmay need to be extended. Sometimes a small part with a center offset hasa sensor height that is smaller than the minimum height in the table.There are also gaps in the data, due to the presence of “guard” regions,as discussed herein.

Data gaps are addressed by a linear interpolation method.

For sensor heights that are smaller than the minimum sensor height inthe sensor height calibration table the table is extrapolated to a zeroheight. The last 10 points in the sensor height calibration table arefit to a line. Then additional points are added to the sensor heightcalibration table between the table's minimum height and a height ofzero.

The same process is carried out to extrapolate the sensor heightcalibration table to the maximum sensor height allowed (0.750″).

Sensor Height Table Extrapolation—Outputs

The outputs of the process are additional sensor height calibrationtable entries, generated from linear extrapolations to zero height andto maximum height.

Sensor Height Table—Zero Height Position

For each sensor calibration table an offset is computed to ensure thatthe sensor height is zero when the sensor views the calibration cone'sbegin cone cylinder 96 (const diam-0 region).

Sensor Height Table—Zero Height Position—Outputs

-   -   sensor height zero offset for (8) sensors

This data meets calibration goal G-2: sensor height zero positionalignment of different sensors.

Thread Signal/Data Processing

Introduction

What follows is a description of the structure of the thread parameterestimation process. This process provides one embodiment of the standardthread measurement “feature” in the system 10.

Thread Signal Processing

Thread signal processing is the process of estimating the followingthread parameters:

-   1) pitch-   2) major diameter-   3) minor diameter-   4) functional diameter-   5) lead deviation-   6) pitch diameter.

The input data to the process is “calibrated part data”. This data setconsists of (8) vectors, one for each photodetector or sensor. Eachvector consists of an indexed table of elements, each containing a (z,h)pair. Each (z,h) pair measures the position of the UUT's shadow ray in acoordinate system that represents each sensor's view of the UUT. z is ameasurement of calibrated sensor stage axis position, and represents thedistance along the stage axis between the current data point and thestage position where the light plane hits the beginning of thecalibration cone. h is a measurement of calibrated sensor height andrepresents the distance between the middle of the beginning cylinder ofthe calibration cone and the shadow ray, perpendicular to the stageaxis.

As the thread signal processing proceeds, a number of intermediate dataproducts are produced in early processing stages that are furtheranalyzed in later stages. These include:

-   -   rough pos/neg crossing locations    -   rough crest locations    -   wire position search intervals    -   left/right flank lines    -   wire positions    -   precise crest/root locations    -   3-crest average/median measurements of major diameter, minor        diameter, pitch diameter    -   3-D crest cylinder axis    -   wire position projections on the 3-D crest cylinder axis    -   3-D crest cylinder diameter    -   3-D crest root-mean-square distance between crest data and fit.

These intermediate data products are analyzed to produce final estimatesof the thread parameters. For example major diameter is estimated astwice the radius of the 3-D crest cylinder. The 3-D crest cylinder axisthen depends on the precise crest/root locations. The crest/rootlocations then depend on the search intervals based on rough crestlocations and pos/neg crossings, and on data from the originalcalibrated part data.

Processing Restrictions

Inspection Region

The thread processing occurs between stage position limits called aninspection region. In the Laser Lab template editor, the user specifiesthe inspection region by manipulating the upper and lower stage positionlimits, overlaid on an image of the part.

These limits utilize the calibrated sensor stage position so thatmeasurements by different lasers are aligned to the approximatelysimilar physical positions on the part.

The estimation of thread parameters is specified to be an averageestimate over all the data within the inspection region. In practice,some of the intermediate data products are estimated outside of theinspection region in order to allow estimation of all thread parameterswithin the full region. For example, a wire position within theinspection region may require a thread crest outside the inspectionregion.

Measurement Assumption for the Inspection Region

The following requirements guide the user's placement of the inspectionregion on the image of the part. At present the analysis software doesnot detect a failure of any of the listed requirements directly.

The first assumption is that the thread parameters be constantthroughout the inspection region. This enables the software to averagethe estimates from different positions within the inspection region andnot be concerned with partitioning or segmenting the data into differentregions for special processing.

This requirement excludes the following types of data from theinspection region:

-   -   the beginning or end of a threaded region, with thread crests        less than full height.    -   a threaded region with a taper.    -   a threaded region with a notch or extensive damage.

A second assumption is that the inspection region contains at least 4-6thread pitches. This amount of data is required to construct several ofthe intermediate data products with the required accuracy. Theintermediate data product that is most closely tied to this requirementis the 3-D peak cylinder described herein.

A third assumption is that the thread be manufactured with a 60-degreeflank angle. Thread processing implicitly utilizes this parameter inseveral places. One of the most direct usages is the conversion of leaddeviation into functional diameter. Other flank angles or other threadform shapes would require different procedures.

A fourth assumption is that the thread has a cylindrical cross section.Non-cylindrical threads would require the 3-D peak cylinder to besuitably generalized. Incorrect fit to a non cylindrical cross sectionwould lead to incorrect lead deviation measures in the currentimplementation.

A fifth assumption is that the thread has a single helix. Currentlydouble threads are not supported.

The software does not check the assumptions. Failure to meet therequirements will typically lead to bias in the thread measurement, orin a failure to successfully measure the inspection region.

In practice these requirements limit the measurement of the followingobjects:

-   -   non-standard thread types, especially self-tapping screws.    -   small threaded regions with 2 or 3 pitches.    -   Taptite trilobe threaded regions.

Rough Crossings

The thread model described hereinbelow is a sampled representation ofone sensor's thread profile, for exactly one pitch. The thread modelstarts at the midpoint of a rising thread flank and ends one pitchlater.

Using a correlation detector the thread model is matched to data withinthe inspection regions, producing thresholded detections within theinspection region, that are called crossings. FIG. 17 shows a sketch ofa thread model matched to the sensor data.

Later processing “refinements” noted herein may make the crossings moreaccurate. The refinements also separate the crossings into positivecrossings (right flank line in FIG. 17) and negative crossings (leftflank line in FIG. 17). FIG. 18 illustrates selected concepts of athread form. The thread model is a lateral sequence of points thatrepresent a best estimate of the outline of one cycle of the threadform.

Rough Crest and Root Positions

A crest/root detector extracts rough crest and root positions betweenthe matched adjacent pairs of positive and negative crossings.

Pitch Estimate

A pitch estimate is required for step set gage wire diameter. Theestimate is required to be accurate enough to unambiguously select aunique gage wire from the set appropriate for the measurement. Thecurrent process utilizes a two-stage process.

This process may be simplified as described herein.

First Estimate

Crossing data is analyzed and averaged over all sensors to create athread pitch estimate, the “crossing pitch”.

Second Pitch Estimate

The steps: set wire gage diameter, wire position search intervals,measure flank lines and measure 3-point diameters noted hereinbelow arecompleted in a first iteration. Then the wire positions are averagedover all sensors and positions to compute a pitch estimate.

Set Gage Wire Diameter

Gage wires are utilized in physical thread measurements of pitchdiameter in the prior art. Two wires are placed in adjacent threads onone side of the UUT, and a single wire is placed on the other side ofthe UUT. A micrometer measures the distance between the reference lineestablished by the two adjacent gage wires and the reference pointestablished by the other gage wire. A tabulated correction formulaconverts the micrometer distance to an estimate of the pitch diameter.

Gage wire sizes are thus selected prior to the thread measurement. To dothis one estimates the thread pitch as previously described and then oneselects the closest gage wire in a set to the pitch estimate. The gagewire set utilized is the one appropriate to the type of measurement;currently there is one set for the metric coarse thread sequence, andanother for a similar English thread set. The gage wire sets are chosenat part template edit time, by making a selection in a pull down list.

Wire Position Search Intervals

One places “virtual” gage wires onto the calibrated sensor datathroughout the inspection region. In order to place the “virtual” gagewires we must identify search intervals for each wire to be located.

A requirement of the following processing steps is that the wirepositions in the inspection region have no gaps. Another requirement isthat a wire position search interval consist of two valid thread crests,one valid thread root between the two thread crests, and validpositive/negative crossings between the crest/root pairs.

One then searches the set of positive/negative crossings and crest/rootpositions for the set of wire position search intervals to analyze. Theresult is a set of intervals, one set per sensor.

Measure Flank Lines

FIG. 19 shows a sketch of a portion of calibrated sensor data in asingle wire position search interval.

The specification of a valid wire position search interval means thatthe form of the calibrated sensor data is approximately as shown in FIG.19. This form was used to create a plan to robustly extract flank linedata.

For the left flank line (example) we analyze all data between the roughpositions of the left crest and the central root. One then determinesthe height limits of a flank line data extraction region that covers 70%(a configurable parameter) of the height interval between left crest andcentral root. This data is extracted into a data set and fit to a line,becoming the left flank line.

The procedure avoids the non-linear regions near the left crest andcentral root. In addition a “flank line valid” flag is computed, basedon the RMS distance between the left flank line and the data within theleft flank line data extraction region. If the RMS distance between theflank line and the data points in the flank line data extractioninterval is larger than 10 μm per point (a configurable parameter), thenthe flag is set to invalid.

The process is repeated for the right flank line and then for all wireposition search intervals.

Measure Wire Positions

The wire positions are calculated, given the left and right flank linesand the wire size. As shown in FIG. 19, the virtual wire is tangent toeach flank line and the resulting position is calculated with a simplegeometric formula.

The position has a valid flag that is true when both flank lines arevalid, and false otherwise. two flank line “valid” flags.

Measure 3-Point Diameters

The 3-point technique is a method to measure the minor, major, and pitchdiameters without explicitly utilizing 3-D information. All computationsare carried out in the 2-D laser sensor coordinate system.

For example, consider the major diameter. It is defined as the diameterof a cylinder that contains all the inspection region's thread crests.

In this method, the top of a thread crest in calibrated sensor (stageposition, height) coordinates forms an elementary measurement. Theelementary measurements are combined into triplets for further analysis.Only crests from the two sensors of a single laser are combined.

Two adjacent thread crest positions in sensor-1 are combined with thethread crest position in sensor-2 that is closest to the averageposition of crests in the first sensor. The two crests in sensor-1 forma reference line. Then the distance from the reference line to the crestin sensor-2 is computed. This is the 3-crest distance for that cresttriplet.

In this manner, the 3-crest distances from all adjacent crest tripletsare computed, for all laser data. The 3-crest distances are all added toa data vector. The 3-crest diameter measurement is either the average orthe median of all the 3-crest distances within the 3-crest data vector.

3-Point Minor Diameter

The 3-point minor diameter computes 3-point distances using precise rootlocations in the sensor data. The 3-point minor diameter is the averageof the 3-point distance vector.

3-Point Major Diameter

The 3-point major diameter computes 3-crest distances using precisecrest locations in the sensor data. The 3-point major diameter is themedian of the 3-point distance vector.

3-Point Wire Pitch Diameter

The 3-point pitch diameter computes 3-point distances using the wirepositions computed in the sensor data. The 3-point wire pitch diameteris the median of the 3-point wire pitch diameter. FIG. 20 is a schematicview which illustrates a 3-point distance method, applied to thread wirepositions. Shown are two wire positions in the top thread form with areference line drawn between them. Also shown is a single wire positionon the bottom thread form with the 3-point distance indicated.

FIGS. 21 and 22 are screen shots from a user interface of a PC whichillustrate intermediate data extracted from a M16×1.5 thread plug gage.FIG. 22 is an enlarged view with its focus on a single thread pitch.

Measure 3-D Crest Cylinder

The measured thread crest position data is analyzed to obtain a 3-Dcylinder with least squares methods. A mathematical description of themethod is given in Appendix C.

The 3-D crest cylinder fit has several output parameters of interest:

-   -   the RMS distance between the crest position data and the fitted        shape.    -   the 3-D location of the cylinder's central axis.    -   the radius of the cylinder.

Project Wire Positions onto 3-D Crest Cylinder Axis

Measured wire positions can be combined with the 3-D location of the 3-Dcrest cylinder's central axis. An imaginary disk, perpendicular to thecylinder axis that goes through the measured wire position marks aposition on the 3d crest cylinder axis.

A data set consisting of the projections of all sensor wire positions isconstructed.

For a perfect helical thread and for perfectly measured wire positionsthe spacing between the positions in the projected wire positions shouldbe exactly P/8, where P is the pitch of the thread. The (8) sensors eachgive a view that is rotated ⅛ revolution between adjacent sensors.

For a right handed thread, the wire positions project onto the axis atincreasing positions in the order L1L, L2L, L3L, L4L, L1R, L2R, L3R,L4R, and then L1L, . . . , etc.

The output intermediate data is a vector, sorted from minimum to maximumsensor stage position of the projected wire positions. In addition eachwire position data item is annotated with labels that specify the laserand sensor that produced the data item and other labels containingadditional information.

Thread Parameter Estimation

Thread parameter estimation utilizes the intermediate data products andmay also correct them based on a model of the measurement, prior toproducing a final thread parameter estimate.

Wire Pitch

Thread pitch is estimated from the wire center intermediate data. Foreach sensor data set the adjacent pairs of wire positions are used tocalculate an adjacent wire pitch, one per adjacent wire positions. Forall lasers, each wire pitch is added to a wire pitch vector.

The wire pitch estimate is the median of the elements in the wire pitchvector.

Major Diameter

Thread major diameter is typically reported as the diameter of the 3-Dcrest cylinder.

If the 3-D crest cylinder fit was unsuccessful, the major diameter isestimated in a different way, detailed below. The cylinder fit can faildue to several factors listed here:

-   -   part inclined at too great an angle with respect to the stage        axis.    -   thread crest positions do not fit a cylinder, the RMS        fit-to-data distance is too large.

When the cylinder fit fails the major diameter is estimated from the3-point major diameter data. This case is special because a previouscondition (cylinder fit) has already failed. We found in practice thatthe cylinder fit most often failed when the threaded region was tooshort or the inspection extended beyond the end of the threaded region.

Because of this bias we found that a simple median of the 3-point majordiameter data would typically be too low, most of the good 3-point datawas concentrated at the highest measurements. In this case the majordiameter estimate is the value such that 20% of the 3-point data ishigher and 80% of the 3-point data is lower.

Calibration Correction

Major diameter is also corrected by a final end-to-end calibration ofthe total system. The reported major diameter is often too low, withbias ranging from −20 μm to 0.

After diameter calibration we expose the system to a set of measuredthread plug gages. One then plots their major diameter bias as afunction of diameter and fit a simple segmented line to the biasresults. These bias fits then are entered into the system configurationfile and are used to correct the measured major diameter with themeasured bias.

Minor Diameter

Thread minor diameter is estimated with the 3-point minor diameterdistance vector. The minor diameter value is the average of the elementsin the distance vector.

Pitch Diameter

Pitch diameter estimation uses two sets of intermediate data products,the wire positions and the 3-D crest cylinder fit.

The pitch diameter estimate calculation is presented in a step-by-steplist below:

a) Compute the pitch diameter contact points with the thread flanks bycalculating the intersection of the wire shape with the left or rightflank lines.

b) Average the left and right points of intersection, and compute thedistance (radius) from the average point to the 3-D crest cylinder fitaxis. This is the pitch diameter radius for each wire position.

c) Calculate the average value of the pitch diameter radius for eachsensor.

d) Correct each sensor's average wire position radius for the partprojection angle, using the angle of the 3-D crest cylinder axis to thestage axis, projected into each sensor's coordinate system.

e) Add left and right sensor corrected pitch diameter radius estimatesto produce an estimate of the pitch diameter for each laser.

f) Average the laser estimates to produce the system pitch diameterestimate.

Correction for Part Projection Angle

The computation of pitch diameter is complicated by projection effects.The laser light performs an almost perfect orthographic (shadow)projection of the thread's shape. However the projection is not the samething as the thread cross section, which is specified in thread designdocuments. The cross section is the thread shape if it were cut by aplane going through the thread's central axis.

The difference is caused by the thread lead angle, which is in the rangeof 1-3 degrees for many typical threads. The lead angle means that thethread cross section is most accurately viewed in shadow when theviewing direction coincides' with the direction of the lead.

It is impossible to position the thread so that a shadow view of thethread is simultaneously aligned with the top and bottom threads. Forthe example of a thread with a 3 degree lead angle, tilting the threadto align the top of the thread with the viewing angle will make theangle between the lead and the viewing angle for the bottom thread about6 degrees.

A correction factor was developed for this effect. If one knows the tiltof the thread with respect to the viewing angle then you can correct theobserved pitch diameter radius for the expected bias caused by theprojection angle. This correction is precomputed and stored in a table.

For each sensor the tilt of the thread with respect to the viewing anglecan be obtained from the 3-D cylinder fit axis. Separate corrections areapplied to the left and right sensors.

Calibration Correction

Pitch diameter is also corrected by a final end-to-end calibration ofthe total system. The reported pitch diameter is often too high, withbias ranging from +5 μm to +35 μm.

After diameter calibration, one exposes the system to a set of measuredthread plug gages. One then plots their pitch diameter bias as afunction of diameter and fit a simple segmented line to the biasresults. These bias fits then are entered into the system calibrationfile and are used to correct the measured pitch diameter with themeasured bias.

Lead Deviation

The lead deviation estimate uses the wire pitch and the locations of thewire positions as projected onto the 3-D cylinder fit axis.

For an ideal helical thread, the wire position projections should resultin a regular pattern along the 3-D cylinder fit axis. The projection ofthe first laser-1, left, wire position should lie about (⅛) pitch fromthe projection of the first laser-2, left, wire position. Lead deviationis the deviation of that pattern from the ideal, measured as a maximumdistance of any projected wire position from the ideal pattern.

The computation of the lead deviation estimate follows a step-by-stepprocedure:

a) Create a wire position projection vector, containing all the data.

b) Sort the wire position projection vector in order of position alongthe 3-D cylinder fit axis.

c) Convert the wire positions of the elements of the vector intodegrees, by multiplying by the factor (360/pitch) and then reducing theelement values modulo 360.

d) Calculate an offset value so that the maximum absolute value of thedegree-valued element positions is minimal. For example with a leaddeviation of 0.010 mm for a 1 mm pitch thread, the absolute value of atleast one degree-value element position would be 3.60 degrees. (0.010mm/1 mm equals (1/100) and 360/100 is 3.60.)

e) Convert the value from degrees to mm. and report as the leaddeviation estimate.

Note that all lead deviation estimates are positive.

Calibration Correction

Errors in measurement mean that the physical measurement of a perfectthread will have a positive lead deviation.

To attempt to correct for this effect, one measures the lead deviationfor a set of thread plug gages and plotted them as a function of gagediameter. The most common form observed is a constant lead deviation of0.010 mm. to 0.020 mm.

This value observed in calibration with thread gages is taken to be abias. This amount of bias is entered into the system calibration fileand used to correct the measured lead deviation for this measurementbias.

Functional Diameter

Functional diameter is currently defined in practice by the fit of aspecial fit gage over the thread. The special fit gage is essentially anut that is split in two by a plane cut through the central axis of thenut. The two halves of the fit gage are held in a fixture that measuresthe distance between the two halves. There is one special fit gage forevery thread type.

Functional diameter is defined as the pitch diameter when the specialfit gage is clamped tightly over a thread plug setting gage. When oneputs a different UUT into the fit gage the fit gage may expand slightly,due to a summation of effects involving the differences between the UUTand the thread plug setting gage used to setup the functional diametermeasurement. The functional diameter measurement is then the thread plugsetting gage's pitch diameter plus the additional separation between thetwo fit gage pieces.

Functional Diameter—Laser Lab Estimator

In the Laser Lab, our functional diameter measurement method is anapproximation of the fit gage method. We do not perform a full 3-Danalog of the physical fit gage. Instead we have made an approximationthat involves the use of lead deviation and the shape of the threadform.

If we imagine the thread form as perfect and also having a 60 degreeflank angle then lead deviations should cause a the thread form fit gagepieces to move apart. A single lead deviation either up or down thethread form axis will cause a single split piece of the fitting gage tomove outward. The amount of outward movement for a 60 degree flank anglewill be equal to (√{square root over (3)}) (lead deviation). Themovement provides a clearance for both positive and negative movementsof the lead, relative to a perfect helical shape.

The Laser Lab estimator for functional diameter, (FD) is given below:FD=PD+√{square root over (3)}(LeadDeviation).Learning the Thread Model

The thread model is a learned, sequence of points that represent a bestestimate of the outline of one cycle of the thread form. The threadmodel is calculated when the inspection region is specified, at templateedit time.

The measure template routine uses a pattern match algorithm with a sinewave pattern to identify periodicity in the inspection region data. Thisprocess determines an approximate thread pitch. The process alsocalculates a starting point in the data vector for the first beginningof the matched pattern, which is an approximation to the first midpointof a right flank line.

With the pitch and the starting point in hand, the measure templateroutine can then calculate an average thread model. Starting with thefirst sample point in the matched pattern, points that are 1,2,3, . . ., N pitches later in the inspection region are averaged to form thefirst point of the thread model. The process is repeated for all therest of the points in the first matched pattern. The thread model isthen stored in the template for later use.

The following is a description of the structure of the trilobe ortrilobular estimation process.

Trilobe Signal Processing

Trilobe signal processing analyzes calibrated part data within theinspection region and produces intermediate data products that areanalyzed by the trilobe parameter estimation process describedhereinbelow. Eight values are produced in trilobe signal processing,four laser-n diameters and four laser-n centers.

Intermediate Data Product Description Laser-n diameter Distance fromleft shadow ray to right shadow ray Laser-n center Midpoint of intervalspanned by left and right shadow rays.

Trilobe Blank Signal Processing

For the trilobe blank, the laser diameter and center are estimated assimple averages of calibrated sensor data within the inspection region.

The laser-n diameter is the average of the mean left sensor height andthe mean right sensor height.

The laser-n center is the difference of the mean right sensor height andthe mean left sensor height.

Trilobe Threaded Region Signal Processing

For the trilobe threaded region, one wants to estimate the parameters ofa trilobe cylinder that touches all the thread crests within thethreaded region.

This process can be subdivided into three parts:

-   -   obtain thread crest locations from thread region global object    -   estimate sensor height from sensor thread crests, robustly    -   compute laser-n diameter and center from sensor heights.

One obtains thread crest locations from the thread region object,keeping only thread crests that are within the inspection region, andthat also have height at least 95% of the median crest height. For avalid inspection region there would then be 5-10 thread crest points persensor for typical usage of the trilobe feature.

To estimate the sensor heights one needs an estimation process that isrobust enough to tolerate several invalid thread crests. A preferredprocess uses a “robust” line fit procedure to obtain a line fit throughthe thread crests that will not be influenced by 1 or 2 invalid crestdata items. Once the “robust” line is found, the sensor height estimateis the “robust” line's height at the midpoint of the inspection region.

Robust Line Fit Procedure

The robust line fit is a simple parameter sampling process. For everypair of points in the data set to be fit, an evaluation line isproduced. A figure of merit for every evaluation line is produced and isthe RMS distance per point between the data and the evaluation line. TheRMS distances are sorted and the evaluation line with the median RMSdistance is chosen.

This procedure is computationally costly but can work correctly with upto 49% of the data as “outliers.”

Potential Issues with Trilobe Region Signal Processing

Inspection Region Taper May Bias Results

The estimation process is model-based and the model is a trilobe“cylinder.” Thus, a taper in the threaded region, such as near a threadpoint, would provide data that the model fitting process would not becapable of analyzing accurately.

Trilobe Threaded Region Crests Should Be Accurately Located

The thread region processing that locates the thread crest input datafor the threaded trilobe estimation process is very general and maymisfit crest shapes that do not match the thread region “crest model.”

Trilobe Parameter Estimation

Trilobe parameter estimation utilizes the intermediate data products,laser-n diameter and laser-n center, to compute the following trilobeparameters:

Trilobe Parameter Description C Radius of circumscribed orcircumscribing circle (V-anvil micrometer maximum diameter) DOrthographic projection diameter (Diameter with standard calipers) ERadius of inscribed or inscribing circle (V-anvil micrometer minimumdiameter) K Trilobe “out of round” parameter (i.e., degree oflobulation) K = E − D = D − C 2K = E − C Angle Rotation angle (between 0and 60 degrees) Angle = 0 when Trilobe flat is parallel to x-axis, andlowest point in y Angle = 60 degrees when trilobe flat is parallel tox-axis, and highest point in y xCenter, The centerline coordinates ofthe trilobe shape. yCenter

“D” Parameter

The trilobe D parameter can be estimated as the average of the laser-ndiameter measurements in the four lasers.

All the values should agree, within the margin of sensor errors.

If one measures a perfect trilobe shape gage, the differences betweenthe laser-n diameters and “D” are diagnostic of measurement accuracy andbias. The RMS distance between the laser-n diameters and “D” is ameasure of diameter measurement uncertainty. The maximum differencebetween “D” and laser-n diameter is a measure of the maximum per sensordiameter measurement bias.

Iterative Computation of K, Angle, xCenter, yCenter Parameters

The computation of the K, Angle, xCenter, and yCenter parameters usesonly the laser-n center intermediate data product. The four laser-ncenter data items are exactly enough items to compute the four unknowntrilobe parameters, there is no redundancy.

A direct four parameter search process is difficult. The search wassimplified to an iterative two parameter search with the followinganalysis.

If one assumes that the (xCenter, yCenter) centerline coordinates of thetrilobe shape are known, one can estimate K, Angle with an exhaustivesearch process, described hereinbelow. Once one has estimates of D, K,and Angle, one has a complete description of the trilobe shape.

With the trilobe shape description one can calculate the differentprojections of the trilobe shape onto the left and right sensors. Withthe left and right sensor projections of the trilobe shape one can usethe laser-n center data to estimate the trilobe centerline coordinates,xCenter and yCenter.

Finally, with the trilobe centerline coordinates one can change theorigin of the coordinate system specifying the laser-n center data sothat the origin of the next set of laser-n center data is at the trilobecenterline coordinate estimate. Then the process is repeated with thetransformed laser-n center data as input. In this process, the K, Anglesearch progress is presented with data that eventually has a centerlinethat is very close to (0,0). At that point, one knows all the trilobeparameters, K, Angle, D, xCenter, and yCenter.

Here is a short description of the process.

Repeat {   Perform K, Angle search.   Determine xCenter, yCenter.  Recenter Laser-n Center coordinate system. } until (centerlinecorrection is very small).

K, Angle Search

The K, Angle search is carried out by exhaustive enumeration. A2-dimensional grid is constructed with 1-dimension being the possiblediscrete values of K in the interval (0 . . . kmax) and the otherdimension being the possible discrete value of angle in the interval (0. . . 60) degrees. At each grid point K, Angle, xCenter, yCenter areused to calculate the laser-n center values that would have producedthose values and then an RMS distance between the calculated and actuallaser-n center values.

In a preferred implementation, the discrete grid is sized 25×25resulting in 625 K, Angle parameter values and 625 RMS values. Theminimum RMS grid value selects the K, Angle output value.

K, Angle Fine Search

The K, Angle search is increased in precision by a subdivided search. Arectangular region of K, Angle space equal to a 2×2 grid in the originalK, Angle discrete grid is subdivided into a 25×25 grid and searched.

Then the process is repeated a second time, subdividing the fine grid inthe same manner.

The result is a more accurate K, Angle calculation at much less costthan a brute force search through a 3906×3906 grid. (The cost is about3× times a 25×25 grid search.)

Determine xCenter, yCenter

Once K and Angle are known a new estimate for the trilobe centerlinecoordinates can be obtained.

(1) Estimate sensor height difference caused by trilobe shape, for allfour lasers. This difference is a function of the difference between thelaser and trilobe shape angles.ΔH(laser, trilobe)=f(K, Angle−laserAngle).

(2) Correct the sensor height difference for the trilobe contribution.ΔH(laser)=ΔH(laser, data)−ΔH(laser, trilobe).

(3) Compute the trilobe centerline coordinates by a least squares fit ofthe corrected sensor height differences.(xCenter, yCenter)=g(ΔH(1), ΔH(2), ΔH(3), ΔH(4)).

Convergence Criteria

One says the iteration converged when the difference between theestimates of K from the current and the previous iteration is less thana predetermined parameter (nominal value 0.0001).

Computation of Derived Parameters

The derived parameters can be computed from the estimated parameters Dand K:C=D−K.E=D+K.Code Implementation Notes

These notes are to guide future improvements in the code.

Trilobe Thread Parameter Estimation—Differences from

Standard Thread Processing

Trilobe region thread estimation has some differences from standardthread processing.

Most of the differences arise from the fact that standard threadprocessing utilizes a cylinder of circular cross section whereas trilobethread processing utilizes a cylinder with a trilobe cross section.

List of Features

Calibrated Sensor Stage Axis Position Zero—by Analysis of Begin ConeEdge Signal

The scanning optical head system described above produces a sampledimage of the amount of light and shadow in a particular sensor's beam. Asample is produced each 4 μm of stage travel. The absolute stageposition is not precise or repeatable, as also discussed.

In order to make the stage position coordinates refer to a commonphysical position, the sensor signal is analyzed to find the position ofthe step edge that marks where the sensor passes the beginning of thecalibration cone 40 at the cylinder 96.

Once the sensor stage positions are all referenced to the common begincone position, the positions of all other features are repeatable tohigh accuracy from scan to scan.

Calibrated Sensor Height Position Zero —by Analysis of Begin Cone CenterPosition

The calibration process that relates sensor digitized raw signals tosensor heights calibrates the relative sensor blockage between the0.125″ cone minimum diameter and the 1.500″ cone maximum diameter.

After the table relating the raw signals to sensor heights isconstructed, the table is used to compute the center of the cone 0.125″beginning cylinder 96. That position is used as an offset to make thecalibrated sensor heights read out zero at the center of the 0.125″ conecylinder 96. This process establishes a common (x,y) center referencecoordinate for each of the (4) light planes.

Calibration Cone Design —Measure 3d Alignment to Stage Axis with

Analysis of Two Constant Diameter Regions

The calibration cone 40 has two regions of 0.750″ diameter that define acylinder in space that is concentric with the calibration cone's centralaxis. By measuring the position of the 0.750″ cylinder as seen by thesensors, the calibration software determines the alignment of the stageaxis 28 and the calibration cone axis.

It is important that the regions measured to define the calibration coneaspect vector have the same diameter. That means that errors in thesensor height calibration have a minimal influence on the accuracy ofthe aspect vector computation.

Calibration Cone Design—Measure Light Plane Angle with “Multi-step”Region

The calibration cone's “multi-step” region contains (5) preciselypositioned mechanical steps. These steps are utilized to compute theroll angle of the light plane with respect to the calibration cone'scentral axis.

Signal processing software measures the precise location of each of the(5) steps. When the light plane has a roll angle with respect to thecalibration cone 40, the difference in position between the steppositions computed from a laser's left and right sensors is proportionalto the sine of the roll angle.

The analysis software utilizes the data from all (5) steps in a leastsquare minimization procedure that computes the roll angle.

Calibration Cone Design—Use Continuous Constant Slope Sensor Height

Calibration Region Allowing Arbitrary Interpolation

Previous experimental designs of calibration cones used stepped edgesfor the purpose of relating raw digitized sensor signals to calibratedsensor heights. If the sensor response varied between the height of twoadjacent steps then the calibration process would not directly measurethe variation and the resulting calibration might make mistakes atintermediate diameters.

The present design provides data at all sensor heights, in the diameterrange 0.125″ to 1.500″.

Light Plane “Layer Cake”—Reduce Cross Talk

As previously mentioned, the (4) laser light planes are arrangedparallel to the bottom plate 61 of the optical head 12, in a regularlyspaced array of heights. Adjacent laser light planes are preferablyseparated by about 2.5 mm. The arrangement is shown schematically inFIG. 10.

This “layer cake” arrangement was chosen specifically to eliminate orreduce “cross talk” between different laser beam lines. For example,light from beam line-1 might scatter from the surface of the UUT and gointo the sensor for beam line-2.

The primary means of interference is due to scattering from cylindersthat are aligned with the stage axis 28, a geometry similar to thegeometry of FIG. 9. (Also see Appendix A.)

When the laser light planes are at different heights, light from laser-2(for example) which is scattered by the UUT, arrives at the sensor forlaser-1 at a height of 2.5 mm relative to the expected light from thelaser-1 light plane. This scattered light can be blocked by a lightplane receiver aperture slit as described in Appendix A with referenceto FIG. 24 (i.e., telecentric apertured stops).

Light Plane Receiver Aperture Slits—Reduce Cross Talk

The light plane receivers 68 each have linear slit apertures, about 1.5mm high, that accept light from its corresponding light plane generator.Each aperture slit is mounted in the optical head 12 at a differentheight, matched to the height of its corresponding light plane. Lightfrom different light plane generators or transmitters 66, scattered bythe UUT, is effectively blocked, thereby increasing measurementaccuracy.

Light Plane Receiver Aperture Pinholes—Reduce Forward Scattered Light

Each light plane receiver 68 includes photodiodes which are each fittedwith circular apertures that make the light plane receiver 68“telecentric”. This aperture pinhole accepts light rays from the nominalangle of incidence and/or from angles of incidence that are onlyslightly different (<1-2 degrees). This means that light beams thatenter the light plane receiver 68 at larger angles of incidence will beblocked by the pinhole mask and not recorded by the measurementcircuitry. Appendix A describes this.

The pinholes reduce systematic measurement errors caused by shinycylindrical parts. For those parts forward scattered light will tend tosystematically reduce the diameter measured because scattered light thatwould be blocked by a rough dark surface finds its way into the lightplane receiver 68.

Light Plane Generator Module—Alignment Method to Ensure Low BeamDivergence

The Laser Lab measurement system 10 has a requirement that the lightrays from each light plane generator module 66 be parallel and notdivergent.

The apparent diameter of a 0.500″ [12.7 mm] cylinder should not changeby more than 0.0001″ [0.0025 mm] as the cylinder center is moved (+/−)0.0394″ [1 mm] from the center of the measurement area. This requirementcouples a required measurement accuracy bias (0.0001″ [0.0025 mm]) withan estimated accuracy of part placement by customers (+/−0.0394″ [1mm].).

This requirement places limits on the alignment accuracy of the lightrays within the light plane, or its divergence. In the worst case, thebeam through the center of the cylinder is at an angle of zero, the leftshadow ray is at an angle of −φ, and the right shadow ray is at an angleof +φ. This would mean the maximum misalignment angle for any shadow rayin the light plane is less than 1.3 mrad.

These maximum misalignment angles translate to an accuracy of focus whenmanufacturing or assembling the light plane generator module 66. Analign and focus instrument or alignment fixture, generally indicated at100 in FIG. 31, provides a high precision mechanical adjustment of theposition of lens 316 (i.e., FIG. 37) so that the beam divergence isminimized when constructing each light plane generator module 66. Oncethe adjustments are complete, the adjustments can be permanently fixedin place by tightening adjustment screws and by gluing the mechanicalattachment points to prevent movement. A fuller description of thealignment method is provided in Appendix D.

Light Plane Generator—Alignment Method to Ensure High Light BeamFlatness

The Laser Lab measurement system 10 also has a requirement that thelight plane be generally flat. It was discovered that if the opticalelements of the light plane generator module 66 were misaligned then thelight plane's image on a flat target would be curved, rather thanstraight.

A curved light plane would make the light plane-to-calibration coneangle calibration described above invalid. A curved light plane wouldalso make the sensor height calibration described above inaccurate. Thecurve in the light plane would make predicting the diameter of thecalibration cone 40 as a function of stage position much less accurateand make the sensor height calibration much less accurate.

The align and focus method as noted above and as described in Appendix Dis designed to allow the light beam flatness of the light planegenerator module 66 to be effectively minimized during moduleproduction. It was found that the angular and rotational alignment ofthe lens 310, 312 and 316 to the module base plane was an importantvariable. These alignments when performed sequentially allow the lightplane generator module 66 to be setup to meet the flatness requirement,at which point the adjustments are permanently fixed in place bytightening adjustment screws and gluing mechanical attachment points toprevent movement.

Flatness is eliminated primarily by the adjustment of the lens 316, FIG.37, the “first cylinder lens.” The align and focus instrument contains arotating arm, with a clamp 188, FIG. 31.

The rotating arm's clamp holds the plate 318, FIG. 37, attached to thefirst cylindrical lens 316, during alignment of the transmitter module.

Rotation of the clamp 188 causes the laser line image at the target 210to transition between line shapes on the target of curved upwards, flat,and curved downwards.

A secondary adjustment is by rotating the lens 316. This adjustmentprimarily affects the inclination of the laser line image at the target210, not the curvature. The inclination is adjusted to make the laserline image horizontal.

The adjustment of lenses 316 and 312 is inter-dependent.

Light Plane Receiver—Alignment Method

The light plane receiver modules 68 also have alignment requirements.The optical elements of the modules 68 are precisely positioned so thatthey precisely focus the light from their respective light planegenerator module 66 within the pinhole apertures of the detectors (i.e.,FIG. 24).

The receiver module 60 accepts the light from all possible light rayswithin the light plane at approximately the same efficiency, so thegenerator/receiver subsystem (i.e., 66 and 68, respectively) will have asmooth light acceptance profile, as a function of distance across thelight plane. This is a requirement from the sensor height calibrationprocess.

The light plane receiver modules 68 and the light plane generatormodules 66 are capable of working together when mounted on the opticalhead base plate 61 at standard hole positions.

The receiver light plane split line is centered within the light plane.The align and focus method allows for the proper assembly and subsequenttesting of the light plane receiver module 68 and its components asdescribed herein.

Referring now to FIGS. 37-44, there is illustrated a common mirror lensmount, generally indicated at 304, of the transmitter modules 66.Preferably, the mount 304 is manufactured from a single piece of lowexpansion al-mag alloy to ensure dimensional stability over a widetemperature range. The mount 304 is precisely machined to ensure thatthe various reference surfaces of the mount 304 are properly positionedwith respect to each other.

FIG. 37 is an exploded perspective view of the mount 304 together withits various supported mirrors 306, a first cylindrical lens 316, asecond cylindrical lens 312 and a third cylindrical lens 310. The mount304 includes an integrally formed lens holder 314 for the secondcylindrical lens 312. The third cylindrical lens 310 is mounted on afront reference surface 311 of the mount 304. The second cylindricallens 312 is held within its holder 314 which is at least partiallydefined by a reference surface 313 as best shown in FIG. 61. The firstcylindrical lens 316 is held within an adjustable mounting assembly,generally indicated at 320, which includes a base plate 319 and a lensmount 318.

A rear one of the mirrors 306 is held within an adjustable mirror mount322 which is mounted at a back reference surface 323 of the mount 304(i.e., FIG. 39). The front one of the mirrors 306 is mounted internallywithin the mount 304 at an internal reference surface 325 as best shownin FIGS. 43 and 39. Comparing FIGS. 39 and 43 (sectional views of themount 304 without and with the mirrors 306 and the lenses 306, 310 and312, respectively), the rear mirror 306 is mounted with respect to theinclined reference surface 323 whereas the internally mounted mirror 306is mounted within the module 304 with reference to the inclinedreference surface 325.

The mirrors 306 are preferably made of BK-7 material whereas the lenses310, 312 and 316 are made of SF-11 material. The lenses 310, 312 and 316are optimized for a laser beam wavelength of 650 nm. Also, the nominalaffective focal lengths for the lenses 310, 312 and 316 are 107 mm, 154mm and 2.75 mm, respectively.

The following sequence of assembly steps for the transmitter module 66are followed, which steps are described in detail in Appendix D:

-   1. Secure lens 316 to lens mount 318 using a UV adhesive-   2. Secure lens 312, lens 310 and front mirror 306 to lens and mirror    mount 304 using UV adhesive-   3. Secure back mirror 306 to adjustable mirror mount 322 using UV    adhesive-   4. In sequence:    -   position rear mirror 306 mounted to adjustable mirror mount 322        within lens and mirror mount 304 so that a reference laser beam        entering the sub-assembly parallel to the mounting base and at a        height of 0.984″±0.004″ exits the mounting aperture at the front        reference surface 311 of the mount 304 at a height of        1.679″±0.020″ and remains parallel to the mounting base within        0.05°    -   position lens 312 so that the reference laser beam deviates less        than 0.03°    -   position lens 316 rotationally perpendicular to the beam axis so        that the beam is visually flat    -   position lens 316 along the beam axis so that the beam        divergence in the horizontal axis is within +0.10 m radians    -   position lens 312 rotationally perpendicular to the mounting        surface of lens and mirror mount 304 so that the beam is        parallel to the assembly mounting base within 0.25°    -   position lens 310 so that the reference laser beam is 1.679″        from the assembly mounting surface when measured at a distance        of 10.55″-   5. Completed assembly specifications    -   beam tilt perpendicular to laser beam axis—less than 0.50°    -   beam tilt parallel to laser beam axis—less than 0.10°    -   beam height—1.679″±0.020″    -   beam divergence—less than 0.25 m radians-   6. Secure all fasteners and adjustable components using epoxy    adhesive

Referring now to FIGS. 45 and 46 a through 46 d, there is illustrated indetail the laser steering mirrors 62 and their associated componentparts illustrated in FIG. 5. The various relative positions of the lasersteering mirrors 62 as they are adjustably mounted at the top surface ofthe base plate 61 are illustrated in FIGS. 46 a, 46 b, 46 c and 46 dwherein each mirror 62 is mounted at a different angle with respect toits respective translation plate 400. In turn, each of the translationplates 400 is adjustably mounted to the plate 61 at its top surface bymounting screws (FIG. 5). Mounting screws 402 and their associatedwashers 403 secure flanges 404 of their mirror mounts, generallyindicated at 406, to their translation plates 400. In turn, the mirrormounts 406 are located on their plates 400 at different angularpositions, as illustrated in FIGS. 46 a through 46 d. Each plate 400includes an elongated aperture 408 which allows the plates 400 to beadjustably positioned at precise angular positions on the top surface ofthe plate 61. Pins 410 are provided for precise mounting the mirrormounts 406 to their plates 400 and mirror mounting plates 412 to theirrespective mirror mounts 406. Screws 414 and their associated washers416 are provided for securing the mount plates 412 to the mirror mounts406. The mirrors 62 are secured to their mount plates 412 by anadhesive.

Referring now to FIG. 47, which is an exploded perspective view of oneof the light plane receiver modules 68. There is illustrated in FIG. 47a mirror and lens mount, generally indicated at 500. The mirror and lensmount 500 is substantially identical to the mirror and lens mount 304 ofeach of the light plane generator modules 66. The receiver module 68also includes a photodetector mount, generally indicated at 502, a lensmount receiver pair, generally indicated at 504, and a cylindrical lens506, which is substantially identical to the cylindrical lens 312 of thelight plane generator module 66. The receiver module 68 further includesa pair of spherical lenses 508 which are fixably secured at a frontsurface of the lens mount 504 in spaced relationship by an adhesive. Thereceiver module 68 still further includes a pair of circular aperturedelements 510 and a detector PCB assembly mount 512 on which a pair ofphotodetectors 514 are mounted in spaced relationship. The lenses 508typically are designed to operate at a wavelength of 650 nanometers, aremade of SF-11 material and have a nominal affective focal length of 25.8mm.

The photodetector mount 502 includes upper and lower halves 501 and 503,respectively, which are secured together by screws 505.

Typically the lenses 508 are secured to the lens mount 504 with a UVadhesive. Then the lens mount 504 is secured to the mount 500 by screws516 and their associated washers 518, as illustrated in FIG. 5.Typically, the lens 506 is adhesively secured to a front referencesurface of the mount 500, as also illustrated in FIG. 5, by followingthe sequence of steps noted below.

The apertured elements 510 are secured within spaced holes in thephotodetector mount 502 so that the elements 510 are intimate with orimmediately adjacent to the supported detectors 514 and centered withinthe mount 500. The following sequence of assembly steps are followedwhich are described in detail in Appendix D:

Position lens 506 at the front reference surface of the mount 500 so areference laser beam entering the subassembly is parallel to a bottomreference surface of the base of the mount 500 at a height of preferably1.679″±0.004″ and deviates less than 0.03°.

Position lens 506 rotationally perpendicular to its mounting surface ofthe mount 500 so that the reference beam frame is parallel to themounting base within 0.25°.

Position the receiver lens pair 508 within the mount 500 along the beamaxis so that the energy as measured using a reference pair of detectorsis balanced within 2%; and

Position by resulting detector assembly so that the energy measuredusing the mounted detectors 514 is balanced within 1%.

After the above-noted steps are performed, all the fasteners andadjusted components are secured using an epoxy adhesive.

The detector PCB assembly mount 512 is secured at the back surface ofthe mount 500 by screws 520 and their respective spacers 522.

Glossary

-   3-Point distance: Distance from a single point in one laser's sensor    to a reference line produced by two points in the laser's other    sensor.-   3-Crest diameter: Diameter measurement produced from a statistical    average or median of 3-point distances.-   Align and focus instrument: An optical/mechanical fixture for    manufacturing or assembling the light plane generator and light    plane receiver modules to the required optical tolerances.-   Base/slide unit: The physical base for the Laser Lab measurement    hardware. The unit includes a relatively large, heavy triangular    base and a vertical unit containing a motor, a slide, and a linear    encoder.-   Beam line: A set of optical, mechanical, and electrical components    contained inside an optical head that create a light plane from a    single laser beam and convert the shadowed light plane to electrical    signals for transmission to the PC tower.-   Calibrated part data: A data structure consisting of calibrated    sensor data from (4) lasers, each having left and right sensors.-   Calibrated sensor data: A data structure containing a vector of    (stage position, height) measurements for one sensor.-   Calibration cone: A precisely manufactured single piece of tool    steel with a “cone-shaped” outline. The outline includes cylindrical    and frustum-shaped outlines. The calibration cone or device is used    within the system to convert sensor digitized raw signals to    physical measurements.-   Calibration data: The set of tables and parameters that are computed    during the calibration analysis process from sensor raw data. This    data set is also an input to the sensor data calibration process    that converts sensor raw data to sensor calibrated height data.-   Correlation detector: A signal processing algorithm that matches a    pattern with a vector data set.-   Diameter calibration: The process of creating calibration data sets    for each sensor. Each data set allows analysis software to convert    the sensor digitized raw signal to a calibrated sensor height. The    process is typically a two stage process including sensor height    calibration and final diameter correction.-   Final diameter correction: A final correction to diameter    measurements based on a system-level diameter gage pin calibration.-   Flank angle: The angle between the thread's flank and the thread's    axis.-   Flank line—data extraction region: The region containing the central    (70%) portion of sensor data for one flank line.-   Inspection region: All the calibrated part data between two stage    position limits, start and end positions.-   Intermediate data: Data produced by signal processing of calibrated    sensor data that is not saved in the template, but utilized in    estimates'of thread parameters.-   Intermediate data—correlation crossing: A match point from the    correlation between a pattern and (typically) sensor height data.-   Intermediate data—flank line: The line fit to data within the flank    line data extraction region.-   Intermediate data—crest/root: Measurements of the locations (stage    pos, height) of the crests and roots present in the thread form.-   Intermediate data—thread 3-D cylinder: Measurement of the 3-D    cylinder formed from the least squares fit of the thread crests. The    cylinder has parameters that include its diameter, per point RMS    distance between data and fit, and cylinder axis.-   Intermediate data—wire position: Location of a virtual wire as    determined from the positions of the left and right adjacent flank    lines.-   Intermediate data—wire position search interval: Region that    contains two adjacent thread crests, a thread root and two thread    flanks.-   Laser number: The enumeration of the (4) lasers in the optical head.    The lasers are numbered laser-1 through laser-4.-   Light plane generator or transmitter module: Physical module with    optical and mechanical components that converts a light beam    generated by a laser into a plane of light having parallel light    rays.-   Light plane receiver module: Physical module with optical,    mechanical, and electrical components that converts a plane of light    having parallel light rays into left and right electrical signals.-   Light plane split line: Imaginary line that splits a light plane    into left and right parts or portions. The line is defined in the    light plane receiver module and represents the response to a shadow    in the light plane in the left/right receivers.-   Left/right receiver (Rcvr): The receiver is a component of the light    plane receiver module that converts light energy incident on its    surface (i.e., image plane) into an electrical signal. The signal    current is basically proportional to the amount of incident light.-   Left/Right sensor digitized raw signal: The output of the light    plane receiver electronics, after processing of the left or right    receiver signals, preferably including current-to-voltage    conversion, amplification, analog filtering, and digitization.-   Light plane receiver electronics: System module that converts left    and right electrical signals from the laser line receiver module    into digitized left and right raw sensor signals, and stores the    results in the PC memory.-   Measurement trigger signal: Signal from linear encoder electronics    transmitted to receiver electronics. One measurement trigger signal    pulse causes all sensor signals to be sampled and stored.-   Optical head: A container such as a sealed metal box containing (4)    beam lines and supporting electronics.-   Part holder: A mechanical subassembly mounted to the base/slide    unit. The subassembly contains the part holder base on which the    parts are received and retained while being scanned. The part holder    also holds the calibration cone in a stable position so it can also    be scanned.-   PC tower: The PC tower is a chassis containing a computer, a set of    control electronics modules, and a number of power supplies.-   Sensor digitized raw signal: The output after processing of a    left/right receiver signal by the light plane receiver electronics.-   Sensor height: The distance from the laser plane split line to the    light/dark shadow edge of a part, measured in the sensor light    plane.-   Sensor height calibration: The process of acquiring and analyzing a    set of data that is used to convert a sensor digitized raw signal to    a sensor height.-   Sensor number: The enumeration of the (8) sensors in the optical    head. The values range from laser-1 left (L1L) through laser-4 right    (L4R).-   Sensor raw data: The set of data generated by the Laser Lab sensor    system during one scan of the calibration cone and the UUT.-   Sensor calibrated height data: The Laser Lab sensor raw data set    generated from one scan, converted to physical units and corrected    for all known issues.-   Shadow ray: The ray of light that just grazes the surface of the    UUT.-   Stage axis: The direction in 3-D space defined by the up/down    movement of the optical head's mechanical stage.-   Stage axis, sensor zero position: The position of the beginning of    the calibration cone, as determined by analysis of the sensor    digitized raw signal.-   Stage axis, calibrated sensor position: The raw position of the    stage axis, corrected to be 0 mm at the sensor zero position. The    calibrated sensor positions are different between sensors, even    between the left and right sensors for the same laser.-   Stage axis, raw position: The stage axis raw position is the value    of the linear encoder counter, maintained by the linear encoder    electronics module. The zero position is set when the stage stops    during a “home” command; the linear encoder electronics module    senses a bottom trip signal, stops the motor, and then zeroes the    encoder counter. This process produces a final resting physical    stage position that varies by several hundred microns, depending on    the length of the move, the stage speed, and other factors. This    physical position is generally too uncertain for direct use in    specifying positions on a part.-   Thread model: An estimate for one cycle of the repeated thread form,    learned at template edit time.-   Thread parameter—functional diameter: Estimate of the diameter of a    virtual nut with the nominal pitch diameter, that could contain the    observed 3-D thread form with all its lead deviations and other    deviations from perfect form.-   Thread parameter—lead deviation: The maximum deviation of the lead    position from the perfect helical form.-   Thread parameter—major diameter: The diameter of a cylinder    enclosing all the thread crests.-   Thread parameter—minor diameter: The diameter of a cylinder through    all the thread roots.-   Thread parameter—pitch diameter: The diameter of a cylinder that    intersects the perfect thread form midway between crest and root.-   Thread parameter—pitch: The average or median axial distance between    adjacent threads.-   Upper tooling: A mechanical subassembly mounted to the base/slide    unit. The upper tooling includes a long stainless steel rod that can    be moved up and down to hold a variety of different-sized parts.    This tooling also includes a spring-loaded part clamp that    facilitates placement, retention and release of parts.-   Unit under test (UUT): The part being measured.

While embodiments of the invention have been illustrated and described,it is not intended that these embodiments illustrate and describe allpossible forms of the invention. Rather, the words used in thespecification are words of description rather than limitation, and it isunderstood that various changes may be made without departing from thespirit and scope of the invention.

Appendix A

Split Laser—Diameter Bias Due to Beam Scattering

SUMMARY

-   -   scattering of light from a reflective cylinder leads to an        underestimate of its diameter.    -   the scattering from a cone section has a large underestimate        than the cylinder.    -   introducing a pin hole aperture in the laser receiver can limit        this effect, by making the laser receiver telecentric, or        sensitive to light from a narrow range of incident angles.        Scattering from a Perfectly Reflective Cylinder

Scattering of light from a cylinder will cause a systematicunderestimate of the cylinder's diameter in the split laser system.Light incident near the surface of the cylinder is scattered at aglancing angle as illustrated in FIG. 23. If the scattered light getsinto the split laser detector, then the cylinder diameter appearssystematically smaller. The effect is largest if the cylinder isperfectly reflective, and is absent if the cylinder is perfectlyabsorbing. The magnitude of this effect is calculated to set a limit onthe systematic size underestimate.

The Scattering Angle

One can imagine a beam of light that hits the cylinder and is reflected.The beam's direction of travel would reach a depth of ΔH within thecylinder if the beam direction were continued on a straight line. Thebeam is deflected by an angle θ_(refl)=2θ by the perfectly reflectingsurface as shown in FIG. 23.

For a cylinder of diameter d the following relationships hold betweenthe scattering angle, θ_(refl)=2θ, the depth ΔH, and the diameter, d.ΔH=d(1−cos θ)ΔH≅d(θ²/2)=d(θ_(refl))²/8θ_(refl≅)229 {square root over (2ΔH/d)}

-   For ΔH=0.0001″, d=0.5000″, then θ≅2√{square root over (0.0004)}=0.04    radian=2.3 degree.-   For ΔH=0.001″, d=0.5000″, then θ≅2√{square root over (0.004)}=0.13    radian=7.2 degree.    Effects on the Diameter Measurement

In the ideal case, with no scattering, the amount of light that reachesthe detector is from line generator light rays that do not intersect thepart. The light signal is then related to the orthographic projection ofthe part, perpendicular to the beam direction.

With scattering, light that would have been blocked may enter thedetector.

The scattered light could fail to enter the detector due to one of thefollowing effects:

-   -   light absorption at the cylinder surface (such as by black        coatings),    -   scattering light may miss the entrance to the laser light        receivers.

One can calculate an upper limit to the underestimate of light blockageby a perfectly reflecting cylinder. It is assumed that all lightscattered through angles smaller than angle θ_(max) will be received inthe laser light receiver. All light scattered through larger angles islost and is not received in the laser light receiver.

The diameter blockage underestimate is:

Diameter Blockage Underestimate ≡ 2ΔH ≅ d(θ_(max))²/4 BlockageUnderestimate θ_(max) Diameter 0.00005″ 0.02 radian 0.5000″ 0.00020″0.04 radian 0.5000″ 0.00080″ 0.08 radian 0.5000″Limiting Diameter Measurement Bias—Using Telecentric Ideas at the LaserReceiver

As illustrated in FIG. 24, the simplest model of the laser receiverincludes a focusing lens for each of the left and right sensors, and alaser diode to measure the laser light at each sensor's focusinglens'focal plane. The laser diodes are placed at the focus for a lightsource at infinite distance.

In this model, when a light ray is incident on the focusing lens at adifferent angle, then the light is focused at a slightly different placein the len's focal plane.

The change in position at the focal plane between light incident alongthe optical axis and light incident at an angle θ, is Δpos=ftan(θ)≈f(θ), where f is the focal length of the lens.

A pinhole aperture is added in front of the laser diode to make thelaser receiver sensitive to light only in a small range of incidentangles. Similar apertures are used in the construction of telecentriclenses. The pinhole aperture size is shown in the following table.

pin hole size = 2fθ_(max) Pinhole Diameter (mm) θ_(max) focal length 2mm 0.01 radian 100 mm  1 mm 0.01 radian 50 mm 2 mm 0.02 radian 50 mm 0.5mm   0.01 radian 25 mm 1 mm 0.02 radian 25 mm 2 mm 0.04 radian 25 mmLight Deflection from a Cone Section

The deflection of light from a reflective cone is also of interest,since conical shapes (i.e., frustums) are utilized in the Laser Labcalibration fixture 40 as illustrated in FIG. 25.

The normal to the cone at the point where the incident beam is justtangent is:{circumflex over (n)}=[−cos(θ_(cone)),+sin(θ_(cone)),0].

The normal to the cone rotated an angle θ_(rot) about the y-axis (seeFIG. 23):{circumflex over(n)}=[−cos(θ_(rot))cos(θ_(cone)),+sin(θ_(cone))−sin(θ_(rot))cos(θ_(cone))]

The incident light's direction of travel is:{circumflex over (l)}=[0,0,1].

The incident light's direction of travel after reflection from a mirroris:

$\begin{matrix}{{\hat{l}}_{refl} = {\hat{l} - {2{\hat{n}\left( {\hat{l} \cdot \hat{n}} \right)}} - {2\;{\hat{n}\left( {\hat{l} \cdot \hat{n}} \right)}}}} \\{= {2\;{\hat{n}\left( {{\sin\left( \theta_{rot} \right)}{\cos\left( \theta_{cone} \right)}} \right.}}} \\{= {2\;{\sin\left( \theta_{rot} \right)}{{\cos\left( \theta_{cone} \right)}\begin{bmatrix}{{{- \cos}\left( \theta_{rot} \right){\cos\left( \theta_{cone} \right)}},{{+ {\sin\left( \theta_{cone} \right)}} -}} \\{{\sin\left( \theta_{rot} \right)}{\cos\left( \theta_{cone} \right)}}\end{bmatrix}}}} \\{= \begin{bmatrix}{{{- 2}\;{\sin\left( \theta_{rot} \right)}{\cos\left( \theta_{rot} \right)}{\cos^{2}\left( \theta_{cone} \right)}},} \\{{{+ 2}\;{\sin\left( \theta_{rot} \right)}{\cos\left( \theta_{cone} \right)}{\sin\left( \theta_{cone} \right)}},} \\{{- 2}\;{\sin^{2}\left( \theta_{rot} \right)}{\cos^{2}\left( \theta_{cone} \right)}}\end{bmatrix}} \\{= \begin{bmatrix}{{{- {\sin\left( {2\;\theta_{rot}} \right)}}{\cos^{2}\left( \theta_{cone} \right)}},{{+ {\sin\left( \theta_{rot} \right)}}{\sin\left( {2\theta_{cone}} \right)}},} \\{{- 2}\;{\sin^{2}\left( \theta_{rot} \right)}{\cos^{2}\left( \theta_{cone} \right)}}\end{bmatrix}} \\{{\hat{l}}_{refl} = {\left\lbrack {0,0,1} \right\rbrack - {2\;{\hat{n}\left( {\hat{l} \cdot \hat{n}} \right)}}}} \\{= \begin{bmatrix}{{{- {\sin\left( {2\;\theta_{rot}} \right)}}{\cos^{2}\left( \theta_{cone} \right)}},{{+ {\sin\left( \theta_{rot} \right)}}{\sin\left( {2\;\theta_{cone}} \right)}},} \\{1 - {2\;{\sin^{2}\left( \theta_{rot} \right)}{\cos^{2}\left( \theta_{cone} \right)}}}\end{bmatrix}} \\{= \begin{bmatrix}{{{- {\sin\left( {2\;\theta_{rot}} \right)}}{\cos^{2}\left( \theta_{cone} \right)}},{{+ {\sin\left( \theta_{rot} \right)}}{\sin\left( {2\;\theta_{cone}} \right)}},{{\cos\left( {2\;\theta_{rot}} \right)} +}} \\{2\;{\sin^{2}\left( \theta_{rot} \right)}{\sin^{2}\left( \theta_{cone} \right)}}\end{bmatrix}}\end{matrix}$

The angle θ_(refl) between incident and reflected light beams can becomputed.cos(θ_(refl))=({circumflex over (l)} _(refl) ·{circumflex over(l)})=cos(2θ_(rot))+2 sin²(θ_(rot))sin²(θ_(cone))

For a cylinder, (θ_(cone)=0), one gets the expected result:θ_(refl)=2θ_(rot)

For small angles scattering angles, (θ_(rot),<<1), and arbitrary coneangles one has:cos(θ_(refl))=1−sin²(θ_(rot))cos²(θ_(cone))1−(θ_(refl))²/2≈1−2(θ_(rot))² cos²(θ_(cone)). θ_(refl)≈2θ_(rot)cos(θ_(cone))Practical Considerations in the Laser Lab System

The Laser Lab cone fixture 40 has cone sections (i.e., frustums) withcone angle θ_(cone)=35 degrees and cos(θ_(cone))=0.820. What this meansin practical terms is that the scattering angle on the cone sections isabout 20% less than the scattering angles on the cylindrical sections.The approximate scaling developed in the previous sections indicatesthat the diameter underestimate would then be about 10% more than forthe straight cylinder case.

Energy Absorption on the Cylinder

If the cylinder surface were coated with a light absorbing coating, thenit might be that the reflected light would continue on in the samedirection as the perfectly reflecting case, but with reduced intensity.

One can develop a model similar to the one noted above. In that modelall light scattered between angles 0 and θ_(max) enters the split laserdetector of FIG. 24. Suppose that a fraction f_(absorb) of the scatteredlight, between angles 0 and θ_(max), is absorbed at the cylindersurface, and a fraction (1−f_(absorb)) continues on to the split laserdetector, then the new model is:2ΔH≈(1−f _(absorb))d(θ_(max))²/4.

Thus, the underestimate is improved by the factor (1−f_(absorb))

Appendix B SUMMARY

This Appendix describes in detail one embodiment of the Laser Labcalibration process.

Table of contents General Parameters 88 Geometrical Parameters of theCalibration 88 Cone Parameters 88 General Parameters of the Calibration89 Cone Signal Processing 89 Full Open Signal Level 89 Rough Edge 90High Precision Edge Positions 91 Laser Roll 91 Data Binning - Cone Slopeor Const Diameter Regions 91 Calibration Model Analysis 92 SensorBlockage Table - Cone Slope Region Data 92 Iteration Control 92Calibration 92 Smooth Cone Description 92 Cone Signal Processing 94 DataPartitioning and Consistency Checks 94 Rough Step Edge Positions 94Rough Slope Edge Positions 95 Rough Edge Position Consistency Checks 95Sensor Data Partition Table 96 Full Open Signal Level 96 Full OpenSignal Level - Consistency Checks 96 High Precision Step Edge Positions96 High Precision Step Edge Positions - Consistency Checks 97 Begin ConeSensor Position Offsets 98 Laser Roll Computation 98 Consistency Checks99 Process Sensor Blockage Data 99 Identify Cone Slope Data Regions 100Data Binning 100 Data Averaging/Feature Generation 100 ConsistencyChecks 101 Process Cone Aspect Angle Data 101 Identify Constant DiameterRegions 101 Data Binning/Averaging/Consistency Checks 101 CalibrationModel Analysis 102 Sensor Blockage Table and Cone Aspect AngleCalibration 102 Cone Projection onto Sensor 103 Computation of SensorBlockage Table 103 Sensor Blockage Table - Interpolation 105 SensorLevel Interpolation Type - “Linear” 105 Computation of Cone Tilt Angle105 Step Edge Position Changes due to Laser Roll Angle 107 Cone 3-DLocation Analysis 107 Travel Z-Axis to Cone Z-Axis Scaling 109 IterationControl 109 Consistency Checks 109 Analysis and Error Propagation 110Effect of Sensor Blockage Table Errors due to Roll Angle 110 on DiameterBias due to Position Offsets Effect of Roll Angle Errors on SensorBlockage Table 111 Effect of Step Edge Location Errors on Roll Angle 111Effect of Multiple Scan Calibration on System Diameter Bias 112 Glossary112General Parameters

Geometrical Parameters of the Calibration

For cone parameters, reference drawing FIG. 4 a.

Sampling Interval 0.004 mm Specified by the encoder. (fixed) Theelectronics generates (8) encoder trigger pulses, one per each sensor,from each encoder pulse.

Cone Parameters

Start Diameter 3.175 mm Diameter of the face at the beginning of thecone. Const Diameter 19.05 mm Diameter of the two const diameterregions. Max Diameter 38.100 mm Diameter of the cone at the first stepedge. Steps 3.810 mm Height of each step. 1.500 mm Width of each step. 5Number of steps. Inclination Angle 34.005 degrees Cone slope angle.

Boundary Positions—Cone Height Model

Distance Height Description 0.000 0.000 Begin cone, bottom of step.0.000 1.587 Begin cone, top of step. Begin const diam-0 region. 3.8101.587 End const diam-0 region. Begin const slope-1 region. 15.578 9.525End const slope-1 region, Begin const diam-1 region. 18.923 9.525 Endconst diam-1 region. Begin const slope-2 region. 33.020 19.050 End constslope-2 region. Begin max diam region. 35.888 19.050 End max diamregion, begin step-1. 35.888 17.145 End step-1. 37.379 17.145 Beginstep-2. 37.379 15.240 End step-2. 38.877 15.240 Begin step-3. 38.87713.335 End step-3. 40.378 13.335 Begin step-4. 40.378 11.430 End step-4.41.877 11.430 Begin step-5. 41.877 9.525 End step-5. Begin const diam-2region. 51.118 9.525 End const diam-2 region.General Parameters of the Calibration

Cone Signal Processing

Full Open Signal Level

500 samples RegionSize 0.4 mm GuardRegion ensure full open signal is notcontaminated by scan startup. 100 samples MaxSignalVariance signalvariance > 0.2% (100/50000) is diagnostic of poor laser sensor signalquality. 45000 MinSignalLevel 64000 MaxSignalLevel. signal levelsoutside this band are diagnostic of laser adjustment problems.Rough Edge

0.4 mm Step Size typical beam “sigma” is 15 samples or 60 um. 0.4 mm isabout 6-7 sigma. 0.4 mm Smooth Size typical beam “sigma” is 15 samplesor 60 um. 0.4 mm covers 6-7 sigma. 200 levels Step Threshold value largeenough to minimize false edge detections. −1500 levels Expected BeginCone Step Height. +1000 levels Expected Standard Step Height. 7000levels Standard Step Height Tolerance. 0.5 Cone Slope Threshold units of(Step Size divided by Sampling Interval), or samples. This is athreshold on a double finite difference, using Step Size. 2 ExpectedCone Slope Height. 3 Cone Slope Height Tolerance. units of (Step Sizedivided by Sampling Interval) or samples. 10 mm Expected Begin ConeOffset. 10 mm Begin Cone Step Position Tolerance. 1 mm Standard StepPosition Tolerance. tolerances are for the relative measurement of stepto begin cone step. 1 mm Standard Slope Position Tolerance. tolerancesare for the relative measurement of step to begin cone step.High Precision Edge Positions

0.1 mm MaxPosDev maximum deviation of position from expected position.3.0 mm MaxHeightDev maximum deviation of step height from expectedvalue. 1000 levels Expected Height. 100 levels MaxRmsDev maximumaveraged RMS(fit-data) for the step edge. 70% LineFitCentralRegionpercentage of central data used for transition line fit. 0.4 mmLineFitRegionSize size of region before and after step used for linefits. 0.4 mm LineFitGuardRegion size of guard region betweenbefore/after step line fit and transition region.Laser Roll

0.100 mm MaxRmsDeltaStepPos step position delta RMS average larger thanthis value is diagnostic of laser beam or calibration device problems.Data Binning—Cone Slope or Const Diameter Regions

0.200 mm GuardRegion minimum distance, diameter data bin to regionboundary. 0.200 mm RegionBinSize size of diameter data bin. 50 levelsMaxDataBinSigma maximum data bin order statistic “sigma” estimator. 25levels MaxDataBinLinDev maximum data bin deviation from neighbor linearestimator. 1 Fit order (0 constant, 1 linear).Calibration Model AnalysisSensor Blockage Table—Cone Slope Region Data

0.01 mm SensorMaxInterpolationError maximum interpolation error, due to“shape” of table. “Linear” SensorInterpolationType specify type ofSensor Blockage Table interpolation. “Linear”ZeroHeightExtrapolationMethod method to extrapolate Sensor Blockagetable. 10 ZeroHeightExtrapolationPoints number of points in theextrapolation data set. “Linear” MaxHeightExtrapolationMethod method toextrapolate Sensor Blockage table. 10 MaxHeightExtrapolationPointsnumber of points in the extrapolation data set.Iteration Control

4 MaxIterations maximum number of Sensor Blockage Table iterations.Calibration

Smooth Cone Description

FIG. 26 shows a schematic outline of the sensor signal produced by thesmooth cone. The different regions of the signal are labeled andexplained below.

FIG. 26: Cone Outline—One Sensor's Signal

The signal is plotted with full open sensor level shown at the bottom ofthe figure and fully blocked sensor level shown at the top of thefigure.

Full Open Region—beginning of scan to beginning of cone.

-   establish the full open level.-   establish the exact position of the beginning of the cone.    Cone Slope Data—two constant slope regions separated by a const    diameter step.-   establish the sensor blockage table, the correspondence between    sensor level and known diameters.    Const Diameter Data—two constant diameter data regions, 0.750″    diameter.-   establish the center line of the cone.-   establish the position of the cone center line, relative to the    stage travel axis.    Steps—Roll Angle Data—five constant height, constant width steps.-   establish the angle of the laser light plane, relative to the cone    center line.    Support Bracket—no data, all light blocked.    Part Support Cylinder—constant diameter fixed support.-   top of this cylinder plus height of part support cap may establish    part base position.    Part Support Cap—constant diameter cap.-   provides base for part.-   center position may vary slightly, caps are not fixed, there are    several for different types of parts, the insertion is easy for    operators to change.-   height and diameter are more precise than the center position.    Cone Signal Processing

In this section the processing and analysis are described that occurprior to creating a sensor blockage calibration table and determiningthe cone tilt angle.

In this section, the raw sensor data is processed and yields features inthe (stage position, sensor level) space.

In the Laser Lab system, the conversion between stage position encodercount and stage position is simple and requires no calibration. Thestage encoder count is multiplied by the Sampling Interval (0.004 mm) toproduce the stage position.

In this section stage positions may be specified interchangeably usingencoder counts or position values.

Data Partitioning and Consistency Checks

The first step is to partition the data into regions for more detailedprocessing. With the smooth cone calibration design the partition can beaccomplished with a combination of the positions of the step edges andknown positions of features on the calibration cone.

Rough Step Edge Positions

The following step edge positions should be identified in the data. Theedges will be identified with a low precision edge finder, usingfinite-difference detection, with smoothing.

Step Description Beginning of Cone Required. Roll Angle Steps 1 through5 ″Rough Slope Edge Positions

The following slope edge (2-nd derivative) positions should beidentified in the data. The edges will be identified with a lowprecision edge finder, using finite difference detection, withsmoothing.

Step Description Beginning of Const Diam-1 Required. End of Const Diam-1″ Beginning of Const Diam-2 Required. End of Const Diam-2 ″Rough Edge Position Consistency Checks

-   c-1: begin cone step edge exists as first step edge near expected    position. “near” has large tolerance, Begin Cone Step Position    Tolerance.-   c-2: required step edges exist near expected positions. Expected    positions are relative to begin cone step edge. “near” has moderate    tolerance, Standard Step Position Tolerance.-   c-2: no other significant edge steps exist in calibration cone    regions. The part region may also have step edges.-   c-3: required slope edges exist near expected positions. Expected    positions are relative to begin cone step edge. “near” has moderate    tolerance, Standard Slope Position Tolerance.

If either of checks 1, 2, 3 fail, the calibration process is stopped anddiagnostic and logging messages are generated on the computer or PC.

Sensor Data Partition Table

The sensor data partition table gives the rough stage position ofboundaries between the calibration data regions, on a per sensor basis.The table is used by downstream functions to provide rough startingpoints for find location modules.

Full Open Signal Level

The full open signal level is computed from data in the full openestimation region, shown in FIG. 27. The region's size is specified aswell as the size of two guard regions, for begin part and for beginscan.

The median and the order statistic corresponding to “sigma” arecomputed, from the sample data within the full open estimation region.

Full Open Signal Level—Consistency Checks

-   c-1: The full open estimation region must fit between the two guard    regions.-   c-2: The “sigma” order statistic must be less than the    MaxSignalVariance parameter.

If these checks fail, the calibration fails and diagnostic and loggingmessages are generated on the computer or PC.

High Precision Step Edge Positions

High precision step edge position processing uses the rough edge steppositions as initial locations to find high precision edge parametersfor 7 step edges. The step edges are the begin part edge, the partsupport cylinder end step, and the 5 steps in the roll angle data. Foreach step edge, 4 parameters are computed, the step position, stepheight, the beam width, and a step quality measure.

The high precision edge detector uses three line fits to the step edgedata, one before the step edge, one after the step edge, and one in thestep edge transition region. A fixed size guard region,LineFitRegionSize, keeps non-linear data out of the before/after stepline fit regions. The central LineFitCentralRegion percent of thetransition region data is used for the transition region line fit. FIG.28 shows these data regions.

The fit degree in the before/after part regions can be adjusted to beeither const (degree-1) or linear (degree-2). For example the begin conestep edge requires before step fit degree-1 and after step fit degree-2.The first step edge requires before step fit degree-2 and after step fitdegree-1. Steps 2 . . . 5 require both before/after step fits to bedegree-1.

For each step a feature is generated, containing:

-   -   transition region midpoint {position, sensor level}=(Up+Lp)/2.    -   beam width estimator=0.4 (Step Size)/(transtion region slope).    -   before/after part line fit parameters.    -   step height=sensor level(Up)−sensor level(Lp).    -   RMS (data-line): per sample value, before part/transition/after        part.    -   RMS (data-line) average=quadrature average of before        part/transition/after part RMS(data-line) values.        High Precision Step Edge Positions—Consistency Checks

-   c-1: all high precision step edges found, and within MaxPosDev    distance of expected positions.

-   c-2: all step edges have heights within MaxHeightDev of expected    heights.

-   c-3: the RMS(data-fit) average is less than MaxRmsDev.

If these checks fail, the calibration fails, diagnostic and loggingmessages are generated on the computer or PC.

Begin Cone Sensor Position Offsets

The set of high precision step edge positions at the begin cone stepedge defines a position offset for each sensor.

The position scale is defined separately for each laser sensor. Theposition offset defines the 0-position for feature processing.

Laser Roll Computation

Laser roll is computed from the high precision step edge positions foundin the roll angle data region.

The measurements and parameters utilized are the following.

StepPos(laser,sensor,i)=high precision step edge position, for i-thstep.

Diam(i)=diameter of i-th step, defined to the midpoint of the step.

The difference between StepPos(laser, Left, i) and StepPos(laser, Right,i)ΔstepPos(laser, i)=StepPos(laser,Right, i)−StepPos(laser,Left, i),is related to the roll angle of the laser line, β. β is positive whenthe Left sensor edge position is less than the Right sensor edgeposition. β would be viewed as a counter clockwise rotation if the leftsensor height is plotted as a positive number and the right sensorheight is plotted as a negative number:ΔstepPos(laser,i)=sin(β)Diam(i).

The 5 equations relating ΔstepPos(laser,i) to Diam(i), can be expressedas the matrix vector equation below:A sin(β)=b.

The data can be reduced with a least squares solution of the matrixvector equation, producing a single parameter estimator, β_(est)

sin (β_(est)) = (A^(T)A)⁻¹(A^(T)b), where ${b = \begin{pmatrix}{\Delta\;{{StepPos}\left( {{laser},1} \right)}} \\\; \\\ldots \\\; \\{\Delta\;{{StepPos}\left( {{laser},5} \right)}}\end{pmatrix}},{A = \begin{pmatrix}{{Diam}(1)} \\\; \\\ldots \\\; \\{{Diam}(5)}\end{pmatrix}}$${A^{T}A} = {\sum\limits_{i}\;\left( {{Diam}(i)} \right)^{2}}$${A^{T}b} = {\sum\limits_{i}\;{\Delta\;{{StepPos}\left( {{laser},i} \right)}{{Diam}(i)}}}$Consistency Checks

-   c-1: Verify that the RMS distance per data point between the data,    {ΔstepPos (laser, i)}, and the values predicted from the fit,    {sin(β_(est))Diam(i)}, is less than MaxDeltaStepPosVariance.    Process Sensor Blockage Data

Raw sensor readings are processed in the cone slope region to produce aset of (sensor level, stage position) features. Typically each featureis based on a small region (10-500 samples) of data. The features areonly generated in regions of valid data, for example they are kept awayfrom step edges by a guard region.

Identify Cone Slope Data Regions

The first cone slope data region lies between the begin cone step edgeand the beginning of a const diameter data angle region. The secondregion starts at the end of the same const diameter cone aspect angleregion and extends to the position of the first roll angle step.

Approximate per sensor boundaries of the cone slope data regions areavailable from the Sensor Data Partition Table, computed in DataPartitioning and Consistency Checks step.

Data Binning

Data is only binned further than GuardRegion counts from the cone sloperegion boundaries, to prevent systematic diameter calibration offsets.

The regions are divided into the number of bins specified by theRegionBinSize and the number of sample positions in the cone sloperegion. If the number of available sample positions is not divided byRegionBinSize, then extra samples are added to the guard regions.

The data bins are not overlapping.

Data Averaging/Feature Generation

Sensor data observations within the bin are processed, forming estimatesof sensor levels and variance within the bin.

Position data within the bin are processed to form a bin positionaverage.

The data within a bin are fit to a first order linear model.

For each data bin, a feature is generated, containing:

-   bin position average.-   bin const, first order linear fit terms.-   bin fit value at bin average position.-   bin RMS(fit-data) estimate.    Consistency Checks-   c-1: Each data bin's RMS(fit-data) estimator should be less than    MaxDataBinSigma.-   c-2: Deviation of each data bin's fit value from the value predicted    by linear estimation using two adjacent bins (left and right) should    be less than MaxDataBinLinDev. The end bins are tested using    extrapolation from two left or two right bins.    Process Cone Aspect Angle Data

The cone's const diameter regions are processed similarly to the coneslope regions.

Identify Constant Diameter Regions

Two constant diameter data regions are processed, see FIG. 27, using theSensor Data Partition Table.

Data Binning/Averaging/Consistency Checks

Data binning, averaging, feature generation, and consistency checks arethe same as for the diameter calibration data regions.

Calibration Model Analysis

Sensor Blockage Table and Cone Aspect Angle Calibration

Cone signal processing produces a laser roll angle estimate and tablesof sensor data at specified stage positions, for cone slope and coneconstant diameter regions.

What is actually required is the projection of the cone 40 onto thelight beam, as a function of the stage position. This projection dependson two angles, the laser roll angle β, and the cone tilt angle α, theangle between the stage travel axis and the cone symmetry axis.

Each of the angles is defined per laser, the full set of laser and coneangles is {B_(i), α_(i)}, where the laser index-i is in the interval (1. . . 4).

The laser roll angle is known, but the cone tilt angle must becalculated. Since the cone tilt angle is small, typically less than1-degree, an iterative process can be successfully defined.

Initially one can assume that cone tilt angle α is equal to zero. Withthis assumption one can use the cone model to generate the expectedprojection of the cone onto the sensor as a function of stage position.The set of cone projections paired with corresponding sensor responsesis used to make the Sensor Blockage Table. After construction, the tablegives the amount of material blocking the sensor, as a function of theexpected sensor response.

The sensor blockage table is then utilized to process the constantdiameter region data, producing an estimate of the cone tilt angle α.

With the cone tilt angle estimate α, the expected cone projections ontoThe sensor and the sensor blockage table are recalculated.

The process is repeated until there is negligible change in the conetilt angle and the sensor blockage table estimates.

Cone Projection onto Sensor

FIG. 29 illustrates cone projection geometry wherein bold lines indicatethe cone outline and wherein:

-   -   H₁ is the radius of the start of the cone.    -   H_(N) is the radius at distance-N along the cone axis.    -   β is the laser roll angle.    -   α is the cone tilt angle.    -   γ is the inclination angle of the cone.    -   Δp is the position offset due to laser roll.    -   δ is the change in radius due to laser roll.

Using the spatial relationships shown in FIG. 29, one can calculate thechange in radius due to laser roll, δ.(ΔH−δ) tan(β)=ΔpΔp tan(γ+α)=δΔH tan(β)=Δp tan(γ+α) tan(β)+ΔpΔp=ΔH tan(β)/(1+tan(γ+α) tan(γ)

Finally, one can obtain the change in projection as a function of thelaser roll angle, the cone inclination angle, and the projected coneaspect angle.

$\delta = {\Delta\; H\;{\tan(\beta)}\left( \frac{\tan\left( {\gamma + \alpha} \right.}{1 + {{\tan\left( {\gamma + \alpha} \right)}{\tan(\beta)}}} \right)}$Computation of Sensor Blockage Table

Signal processing of the cone slope data regions produces a list offeatures, one for each sensor data bin. The feature specifies the stageposition and the average sensor level within the data bin:{SensorBlockage_(i)}={StagePosition_(i), SensorLevel_(i)}.

The {SensorBlockage_(i)} feature is processed to create a calibratedsensor blockage feature:{CalibratedSensorBlockage_(i)}={SensorLevel_(i),StagePositionOffset_(i), SensorHeight_(i)}.

The StagePosition of the SensorBlockage data bin and the StagePositionof the begin cone step edge are used to compute the stage positionoffset from the beginning of the cone:StagePositionOffset_(i)=StagePosition_(i)−StagePosition(BeginCone).

Then the height difference between the begin cone step edge and thestage position is computed:Δhi=ConeHeight(StagePositionOffset_(i))−ConeHeight(BeginCone).

The correction to ΔH_(i) uses estimates of angles α, β and γ to computethe change in the cone projection, as seen by the sensor:δ_(i)=δ_(i)(ΔH _(i), α, β, γ), SensorHeight_(i) =ΔH _(i)−δ_(i).

The following steps summarize the computation:

-   sort {SensorBlockage_(i)} features by increasing StagePosition.-   compute {CalibratedSensorBlockage_(i)}.-   install the {CalibratedSensorBlockage_(i)} features in the Sensor    Blockage Table.-   install min,max SensorLevel limits in Sensor Blockage Table.-   compute interpolation error estimates.    Sensor Blockage Table—Interpolation

The Sensor Blockage Table is analyzed by an interpolating function. Theinterpolating function calculates CalibratedSensorBlockage as a functionof SensorLevel.

Min,max SensorLevel limits are utilized to specify the region of validinput for the interpolation function.

Configuration file parameter SensorLevelInterpolationType is utilized toselect different interpolation methods at run time.

Sensor Level Interpolation Type—“Linear”

Linear nearest neighbor interpolation.

The interpolation error estimate compares the sensor height,SensorHeight_(i)=Sh_(i) at stage position offset,StagePositionOffset_(i)=SP_(i), with the interpolated sensor heightderived from positions (i+1) and (i−1).InterpolationErrorEst_(i)=(SH _(i+1) −SH _(i−1))(SP _(i) −SP _(i−1))/(SP_(i+1) Sp _(i−1))−SH _(i), InterpolationErrorEst_(i)=ηSH _(i+1)+(1−η)SH_(i−1) −SH _(i), where η=(SP _(i) −Sp _(i−1))/SP _(i+1) −Sp _(i−1)).Computation of Cone Tilt Angle

Signal processing of the cone's constant diameter data regions producesa list of features, one for each sensor data bin. The feature specifiesthe stage position and the average sensor level within the data bin:{SensorBlockage_(i)}={StagePosition_(i), SensorLevel_(i)].

The Sensor Blockage Table is used to compute features from SB features.

The calibrated constant data region features are utilized to compute alinear fit to the sensor height data as a function ofStagePositionOffset_(i)=SP_(i).

Left and Right sensor data are fit simultaneously for each laser, sincethe cone tilt angle α affects both. Tilt angle α positive causes thesensor heights to increase in the Left sensor and decrease in the Rightsensor.

${\begin{pmatrix}{+ {SP}_{1}^{L}} & 1 & 0 \\\ldots & \ldots & \ldots \\{+ {SP}_{NL}^{L}} & 1 & 0 \\{- {SP}_{1}^{R}} & 0 & 1 \\\ldots & \ldots & \ldots \\{- {SP}_{NR}^{R}} & 0 & 1\end{pmatrix}\begin{pmatrix}{\tan(\alpha)} \\{Offset}^{L} \\{Offfset}^{R}\end{pmatrix}} = \begin{pmatrix}{SH}_{1}^{L} \\\ldots \\{SH}_{NL}^{L} \\{SH}_{1}^{R} \\\ldots \\{SH}_{NR}^{R}\end{pmatrix}$This equation can be expressed in matrix form:

${A\begin{pmatrix}{\tan(\alpha)} \\{Offset}^{L} \\{Offset}^{R}\end{pmatrix}} = {b.}$The least squares solution is:

$\begin{pmatrix}{\tan(\alpha)} \\{Offset}^{L} \\{Offset}^{R}\end{pmatrix} = {\left( {A^{T}A} \right)^{- 1}A^{T}{b.}}$The A^(T)A and Atb have a simple form:

${A^{T}A} = \begin{pmatrix}{\left\lbrack {{\sum\limits_{j}\;\left( {SP}_{j}^{L} \right)^{2}} + {\sum\limits_{k}\;\left( {SP}_{k}^{R} \right)^{2}}} \right\rbrack + {\sum\limits_{j}\;{SP}_{j}^{L}} - {\sum\limits_{k}\;{SP}_{k}^{R}}} \\{+ {\sum\limits_{j}\;{{{SP}_{j}^{L}\mspace{45mu}\left\lbrack {N_{L} = {\sum\limits_{j}\; 1}} \right\rbrack}{\mspace{11mu}\mspace{65mu}}0}}} \\{\mspace{59mu}{- {\sum\limits_{k}\;{{SP}_{k}^{R}{0\mspace{85mu}\left\lbrack {N_{R} = {\sum\limits_{k}\; 1}} \right\rbrack}}}}}\end{pmatrix}$ ${A^{T}b} = \begin{pmatrix}{{+ {\sum\limits_{j}\;\left( {{SP}_{j}^{L}{SH}_{j}^{L}} \right)}} - {\sum\limits_{k}\;\left( {{SP}_{k}^{R}{SH}_{k}^{R}} \right)}} \\{+ {\sum\limits_{j}\;{SH}_{j}^{L}}} \\{+ {\sum\limits_{k}\;{SH}_{k}^{R}}}\end{pmatrix}$Step Edge Position Changes due to Laser Roll Angle

The laser roll angle widens the step profile, and also biases thepositon.

Cone 3-D Location Analysis

The 3-D cone direction unit vector, {right arrow over (α)}, is observedin each of the 4 laser systems as the cone tilt angle. The projection ofthe cone unit vector, {right arrow over (α)}, into the laser system “I”is (α_(ix), α_(iy), α_(iz))

In FIG. 30, α₁, α₂, α₃, α₄ are the projections of {right arrow over (α)}on the y′-axis for the θ=22.5, 67.5, 112.5, 157.5 degree laser sensorsystems. The positive y′-axis is the Right sensor and the negativey′-axis is the Left sensor direction.

The x,y components of the cone unit vector, {right arrow over (α)}, areprojected into laser system “i” with the following equation (the zcomponent along the stage axis is unchanged):

$\begin{pmatrix}\alpha_{nx} \\\alpha_{ny}\end{pmatrix} = {\begin{pmatrix}{{+ \cos}\;\theta_{n}} & {{+ \sin}\;\theta_{n}} \\{{- \sin}\;\theta_{n}} & {{- \cos}\;\theta_{n}}\end{pmatrix}\begin{pmatrix}\alpha_{x} \\\alpha_{y}\end{pmatrix}}$

The “y” relation is the only one used, since one only measures theα_(iy), α_(iz) components, the sensor does not measure α_(ix).α_(ny)=−α_(x) sin θ_(n)+α_(y) cos θ_(n).

Using measurements in all 4 laser systems one can solve for the twocomponents (α_(x), α_(y)) of the cone unit vector, {right arrow over(α)}, by solving the following linear equation:

$\begin{pmatrix}\alpha_{1\; y} \\\alpha_{2\; y} \\\alpha_{3\; y} \\\alpha_{4\; y}\end{pmatrix} = {\begin{pmatrix}{{- \sin}\;\theta_{1}} & {{+ \cos}\;\theta_{1}} \\{{- \sin}\;\theta_{2}} & {{+ \cos}\;\theta_{2}} \\{{- \sin}\;\theta_{3}} & {{+ \cos}\;\theta_{3}} \\{{- \sin}\;\theta_{4}} & {{+ \cos}\;\theta_{4}}\end{pmatrix}{\begin{pmatrix}\alpha_{x} \\\alpha_{y}\end{pmatrix}.}}$

These equations can be solved by the least squares method:

$\begin{pmatrix}\alpha_{x} \\\alpha_{y}\end{pmatrix} = {\left( {A^{T}A} \right)^{- 1}A^{T}b}$ where${A^{T}A} = \begin{pmatrix}{+ {\sum\limits_{i}\;\left( {\sin\;\theta_{i}} \right)^{2}}} & {- {\sum\limits_{i}\;{\sin\;\theta_{i}\cos\;\theta_{i}}}} \\{- {\sum\limits_{i}\;{\sin\;\theta_{i}\cos\;\theta_{i}}}} & {+ {\sum\limits_{i}\;\left( {\cos\;\theta_{i}} \right)^{2}}}\end{pmatrix}$ ${A^{T}b} = \begin{pmatrix}{- {\sum\limits_{i}\;{\alpha_{iy}\sin\;\theta_{i}}}} \\{+ {\sum\limits_{i}\;{\alpha_{iy}\cos\;\theta_{i}}}}\end{pmatrix}$Travel Z-Axis to Cone Z-Axis Scaling

The distance along the travel axis is different from the distance alongthe cone axis, to second order in the angle between the cone axis andthe travel axis.

${\Delta\;{{pos}\left( {{cone} - {axis}} \right)}} = \frac{\Delta\;{{pos}\left( {{travel} - {axis}} \right)}}{\left( {\alpha_{z} = \sqrt{1 - \alpha_{x}^{2} - \alpha_{y}^{2}}} \right)}$Iteration Control

Calibration model analysis is an iterative process. The number ofiterations computed is MaxIterations. After computing the last iterationthe Iteration Control stopping criteria is evaluated to determine if avalid Sensor Blockage Table was constructed. (See consistency checks c-3and c-4 below.)

Consistency Checks

-   c-1: Sorting {SensorBlockage_(i)} features by increasing    StagePosition should produce a SensorLevel list sorted in decreasing    order. This ensures that one can produce a Sensor Blockage Table    that predicts a unique sensor blocking height at every valid    SensorLevel.-   c-2: The maximum absolute value of the interpolation error is less    than SensorLevelMaxInterpolationError.-   c-3: The number of iterations is less than or equal to    MaxIterations.-   c-4: The change in Sensor Blockage Table projection between the last    two iterations Δδ_(i)=δ_(i) ^(n)−δ_(i) ^((n−1)) is less than    MaxSensorProjectionChange.-   c-5: The maximum absolute deviation of {SensorHeight_(i)} from the    value predicted in the cone angle fit is less than MaxRmsHeightDev.-   c-6: The maximum absolute deviation of the four cone tilt angles    {α_(i)} from values predicted by the fit to the 3-D unit vector    {right arrow over (α)} is less than MaxRmsConeAngleDev.    Analysis and Error Propagation

In this section one can see how known errors in the elementary dataitems, such as step edge positions and median sensor values, affect thesystem measurements.

Effect of Sensor Blockage Table Errors due to Roll Angle on DiameterBias due to Position Offsets

Suppose that during calibration there was a roll angle error. Then theleft sensor actual height would be overestimated and the right sensoractual height would be underestimated and systematic diametermeasurement errors would occur. In this situation, the placement offsetof the center of a cylindrical object from the center of the calibrationaxis would cause a systematic offset in the measured diameter.

Roll angle that is too large causes a underestimate of the correctprojection of the cone for the left sensor and an overestimate for theright sensor.H _(L) =H _(L) ⁰(1−ε)H _(R) =H _(R) ⁰(1+ε)

The measured diameter can then be shown to have a systematic offset thatis proportional to the roll angle error.D=H _(L) +H _(R) =H _(L) ⁰(1−ε)+H _(R) ⁰(1−ε)D=(H _(L) ⁰ +H _(R) ⁰)−(H _(L) ⁰ −H _(R) ⁰)εD=D⁰+ε(CtrPosition), where the center position is defined asCtrPosition=(H_(L) ⁰−H_(R) ⁰).

The relative error in the diameter measurement, due to the roll angleerror is:

$\frac{\delta\; D}{D} = {{\delta\left( {\tan(\beta)} \right)}{\tan(\gamma)}{\left( \frac{CtrPosition}{D} \right).}}$Effect of Roll Angle Errors on Sensor Blockage Table

Roll angle errors couple with the cone slope to systematically (example)overestimate the left sensor actual height and underestimate the rightsensor actual height, in the Sensor Blockage Table.H _(L) =H _(L) ⁰(1−tan(β)tan(γ))

For a small roll angle:δH=±H ⁰ tan(γ)δ(tan(β))For H=0.500″, γ=35⁰, δH=0.0001″, one should have δβ≦0.3 mrad=0.016degree.Effect of Step Edge Location Errors on Roll Angle

A simple method to find the roll angle finds the position of two stepedges, and computes the angle from the difference in step positions.tan(β)=(ΔStepPos)/Diameterδ(tan(β)=δ(ΔStepPos)/Diameter

The difference in two uncorrelated step positions has approximately 40%greater uncertainty than a single step position.δ(tan(β))=δ(StepPos)√{square root over (2)}/Diameter

Step positions have σ≈0.005 mm, and at a Diameter=1.400″, this works outto σ(tan(β)≈0.2 mrad.

Effect of Multiple Scan Calibration on System Diameter Bias

The system diameter bias has an error distribution that is similar tothe repeatability distribution for a diameter measurement.

Multiple scans for calibration reduces the bias.

Glossary

Cone tilt angles {α_(i)}: 3-D cone tilt angle projected into eachlaser's coordinate system. 3-D cone tilt angle {right arrow over (α)}:Symmetry axis of cone relative to stage travel axis. Laser roll angle β:Angle of laser light plane, relative to cone symmetry axis. SensorBlockage Table: Table giving size of object blocking the sensor, as afunction of the raw sensor signal.

Appendix C

In general, this appendix describes how to fit a cylinder to a set ofpoints. The set of points could be determined in any manner. Theapplication within Laser Lab is the fitting of a cylinder to the set of“thread crest” locations. This cylinder is used to estimate the threadregion's major diameter. The data for a thread crest cylindermeasurement is a set of (stage z coordinate, sensor height) pairs. Thesedata points are the observed locations of the thread crest. For a6-pitch thread inspection region, the number of data points per threadis 4 (lasers)*2 (sensors)*2 (flanks)*6 (pitches)=96 (data points).

One would then like to fit all the data points to a simple linear modelof the thread crest cylinder in space with 5 free parameters:

-   the cylinder radius, R.-   the slope of the cylinder center line, in x and y, (a_(x), a_(y)).-   the (x,y) center of the cylinder center line, at the beginning of    the inspection interval, (b_(x), b_(y)).

The 8 data sets one has to work with are:

-   {(z_(i)(l, s), h_(i)(l, s))}, where-   l=laser label, varying from 1 . . . 4,-   s=sensor label, L or R, and-   i=data index, varying from 1 . . . N(l,s).

For example, the data set for the laser-2, right sensor is {(z_(i)(2,R), h_(i)(2, R))}.

To fit one laser's data, one can develop a linear matrix equation. Theparameters are:

-   a(l)=slope of cylinder line in laser−l coordinates.-   b(l)=intercept of cylinder line in laser−l coordinates.-   r=radius of cylinder.

${{Z(l)}\begin{pmatrix}{a(l)} \\{b(l)}\end{pmatrix}} = {{\begin{pmatrix}{z_{1}\left( {l,L} \right)} & 1 \\\; & \ldots \\{z_{N{({l,L})}}\left( {l,L} \right)} & 1 \\{z_{1}\left( {l,R} \right)} & 1 \\\; & \ldots \\{z_{N{({l,R})}}\left( {l,R} \right)} & 1\end{pmatrix}\begin{pmatrix}{a(l)} \\{b(l)}\end{pmatrix}} = {\begin{pmatrix}{h_{1}\left( {l,L} \right)} \\\ldots \\{h_{N{({l,L})}}\left( {l,L} \right)} \\{h_{1}\left( {l,R} \right)} \\\ldots \\{h_{N{({l,R})}}\left( {l,R} \right)}\end{pmatrix} + {\begin{pmatrix}1 \\\ldots \\1 \\{- 1} \\\ldots \\{- 1}\end{pmatrix}r}}}$

The one laser equation can be expressed in block form.

${{Z(l)}\begin{pmatrix}{a(l)} \\{b(l)}\end{pmatrix}} = {{h(l)} + {{s(l)}{r.}}}$

The new vector s(l), is a column of +/−1 values, with +1 for the leftsensor values and −1 for the right sensor values.

${s(l)} = \begin{pmatrix}1_{1} \\\ldots \\1_{N{({l,L})}} \\{- 1_{1}} \\\ldots \\{- 1_{N{({l,R})}}}\end{pmatrix}$

One can develop a block matrix equation for a fit of all 4 lasers' datato 9 parameters. The parameters are:

-   a(1), a(2), a(3), a(4),-   b(1), b(2), b(3), b(4),-   r.

The radius parameter is shared by all 4 lasers, reducing the parametercount to 9 from 12.

${Z\begin{pmatrix}\begin{matrix}{a(1)} \\{b(1)} \\{a(2)} \\{b(2)} \\{a(3)} \\{b(3)} \\{a(4)} \\{b(4)}\end{matrix} \\r\end{pmatrix}} = {{\begin{pmatrix}{Z(1)} & 0 & 0 & 0 & {s(1)} \\0 & {Z(2)} & 0 & 0 & {s(2)} \\0 & 0 & {Z(3)} & 0 & {s(3)} \\0 & 0 & 0 & {Z(4)} & {s(4)}\end{pmatrix}\begin{pmatrix}\begin{matrix}{a(1)} \\{b(1)} \\{a(2)} \\{b(2)} \\{a(3)} \\{b(3)} \\{a(4)} \\{b(4)}\end{matrix} \\r\end{pmatrix}} = {\begin{pmatrix}{h(1)} \\{h(2)} \\{h(3)} \\{h(4)}\end{pmatrix} = H}}$

-   Note: Z specifies the matrix containing data from all 4 lasers, and    Z(l) specifies the data matrix containing data from just one laser.

The new 4 laser equation can be solved by standard least squarestechniques. We will show the solution to develop the structure of theZ^(T) Z and Z^(T) H matrices. We don't actually solve for this set of 9parameters in practice. In the next section, we will transform theequation to eliminate the dependencies among the a(l) and b(l)parameters, and reduce the number of unknown parameters to 5.

Z(a(1)  b(1)  a(2)  b(2)  a(3)  b(3)  a(4)  b(4)  r)^(T) = H(a(1)  b(1)  a(2)  b(2)  a(3)  b(3)  a(4)  b(4)  r)^(T) = (Z^(T)Z)⁻¹(Z^(T)H)$\left( {Z^{T}Z} \right) = \begin{pmatrix}{{Z^{T}(1)}{Z(1)}} & 0 & 0 & 0 & {{Z^{T}(1)}{s(1)}} \\0 & {{Z^{T}(2)}{Z(2)}} & 0 & 0 & {{Z^{T}(2)}{s(2)}} \\0 & 0 & {{Z^{T}(3)}{Z(3)}} & 0 & {{Z^{T}(3)}{s(3)}} \\0 & 0 & 0 & {{Z^{T}(4)}{Z(4)}} & {{Z^{T}(4)}{s(4)}} \\\left( {{Z^{T}(1)}{s(1)}} \right)^{T} & \left( {{Z^{T}(2)}{s(2)}} \right)^{T} & \left( {{Z^{T}(3)}{s(3)}} \right)^{T} & \left( {{Z^{T}(4)}{s(4)}} \right)^{T} & {\sum\limits_{l = 1}^{4}\;{{s^{T}(l)}{s(l)}}}\end{pmatrix}$ ${Z^{T}H} = \begin{pmatrix}{{Z^{T}(1)}{h(1)}} \\{{Z^{T}(2)}{h(2)}} \\{{Z^{T}(3)}{h(3)}} \\{{Z^{T}(4)}{h(3)}} \\{\sum\limits_{l = 1}^{4}\;{{s^{T}(l)}{h(l)}}}\end{pmatrix}$

One now determines the numerical results for the important submatrices.

First, the Z^(T)(l)Z(l) submatrix for laser l.

${{Z^{T}(l)}{Z(l)}} = \begin{pmatrix}{{\sum\limits_{i = 1}^{N{({l,L})}}\;{z_{i}\left( {l,L} \right)}^{2}} + {\sum\limits_{i = 1}^{N{({l,R})}}\;{z_{i}\left( {l,R} \right)}^{2}}} & {{\sum\limits_{i = 1}^{N{({l,L})}}\;{z_{i}\left( {l,L} \right)}} + {\sum\limits_{i = 1}^{N{({l,R})}}\;{z_{i}\left( {l,R} \right)}}} \\{{\sum\limits_{i = 1}^{N{({l,L})}}\;{z_{i}\left( {l,L} \right)}} + {\sum\limits_{i = 1}^{N{({l,R})}}\;{z_{i}\left( {l,R} \right)}}} & {{N\left( {l,L} \right)} + {N\left( {l,R} \right)}}\end{pmatrix}$

The result is quite simple, a matrix containing the accumulated sensorpositions, the accumulated sensor positions squared, and the number ofmeasurements for laser l. For this case, you don't have to sumseparately for L and R sensors.

${{Z^{T}(l)}{Z(l)}} = \begin{pmatrix}{\sum\limits_{i}\;\left( z_{i} \right)^{2}} & {\sum\limits_{i}\; z_{i}} \\{\sum\limits_{i}\; z_{i}} & {N(l)}\end{pmatrix}$

Second, the Z^(T)(l)i(l) vector for laser-1.

${{Z^{T}(l)}{s(l)}} = \begin{pmatrix}{{\sum\limits_{i = 1}^{N{({l,L})}}\;{z_{i}\left( {l,L} \right)}} - {\sum\limits_{j = 1}^{N{({l,R})}}\;{z_{j}\left( {l,R} \right)}}} \\{{N\left( {l,L} \right)} - {N\left( {l,R} \right)}}\end{pmatrix}$

This is also simple, containing the difference between left and rightpositions and the difference between the number of left and right dataitems. Separate L and R sensor sums are required here.

Third, the vector Z^(T)(l)h(l).

This vector contains the correlation between heights and stagecoordinates and the accumulated sum of sensor heights.

${{Z^{T}(l)}{h(l)}} = \begin{pmatrix}{{\sum\limits_{i = 1}^{N{({l,L})}}\;{{h_{i}\left( {l,L} \right)}{z_{i}\left( {l,L} \right)}}} + {\sum\limits_{i = 1}^{N{({l,R})}}\;{{h_{i}\left( {l,R} \right)}{z_{i}\left( {l,R} \right)}}}} \\{{\sum\limits_{i = 1}^{N{({l,L})}}\;{h_{i}\left( {l,L} \right)}} + {\sum\limits_{i = 1}^{N{({l,R})}}\;{h_{i}\left( {l,R} \right)}}}\end{pmatrix}$${Fourth},{{the}\mspace{14mu}{value}\mspace{14mu}{\sum\limits_{l = 1}^{4}\;{{s^{T}(l)}{{h(l)}.}}}}$

This is the accumulated height difference between L and R sensors.

${\sum\limits_{l = 1}^{4}\;{{s^{T}(l)}{h(l)}}} = {\sum\limits_{l = 1}^{4}\;\left\{ {{\sum\limits_{i = 1}^{N{({l,L})}}\;{h_{i}\left( {l,L} \right)}} - {\sum\limits_{i = 1}^{N{({l,R})}}\;{h_{i}\left( {l,R} \right)}}} \right\}}$${Finally},{{the}\mspace{14mu}{value}\mspace{14mu}{\sum\limits_{l = 1}^{4}\;{{s^{T}(l)}{{s(l)}.}}}}$

This is the difference between the number of left and right data points.

${\sum\limits_{l = 1}^{4}\;{{s^{T}(l)}{s(l)}}} = {{\sum\limits_{l = 1}^{4}\;\left\{ {{N\left( {l,R} \right)} + {N\left( {l,L} \right)}} \right\}} = {{{N(L)} + {N(R)}} = N}}$Implementation—Unique Accumulated Values

A look back at the required terms shows there are only 5 different termsthat each need to be accumulated for each set of laser and sensorindices. This makes a total of 40 unique values.

${{sSqZ}\left( {l,s} \right)} = {{\sum\limits_{i = 1}^{N{({l,s})}}\;{\left( {z_{i}\left( {l,s} \right)} \right)^{2}.{{sZ}\left( {l,s} \right)}}} = {{\sum\limits_{i = 1}^{N{({l,s})}}\;{{z_{i}\left( {l,s} \right)}.{N\left( {l,s} \right)}}} = {\sum\limits_{i = 1}^{N{({l,s})}}\; 1.}}}$${{sHZ}\left( {l,s} \right)} = {{\sum\limits_{i = 1}^{N{({l,s})}}\;{{h_{i}\left( {l,s} \right)}{{z_{i}\left( {l,s} \right)}.{{sH}\left( {l,s} \right)}}}} = {\sum\limits_{i = 1}^{N{({l,s})}}\;{{h_{i}\left( {l,s} \right)}.}}}$

Sums and differences of terms can be expressed in a simple notation.

If the sensor argument is suppressed, then the sum of L and R sensors isindicated as:sSqZ(l)=sSqZ(l, L)+sSqZ(l, R).

If the laser argument is suppressed, then the sum over all lasers isindicated as:

${{{sH}(L)} - {{sH}(R)}} = {\sum\limits_{l = 1}^{4}\;{\left\{ {{{sH}\left( {l,L} \right)} - {{sH}\left( {l,R} \right)}} \right\}.}}$

The symbol Δ indicates a left sensor minus right sensor difference as:ΔN(l)=N(l, L)−N(l, R), ΔN=N(L)−N(R), ΔsZ(l)=sZ(l, L)−sZ(l, R).

With these accumulated values

${{{Z^{T}(l)}{Z(l)}} = \begin{pmatrix}{{sSqZ}(l)} & {{sZ}(l)} \\{{sZ}(l)} & {N(l)}\end{pmatrix}},{{{Z^{T}(l)}{s(l)}} = {\begin{pmatrix}{{{sZ}\left( {l,L} \right)} - {{sZ}\left( {l,R} \right)}} \\{{N\left( {l,L} \right)} - {N\left( {l,R} \right)}}\end{pmatrix} = \begin{pmatrix}{\Delta\;{{sZ}(l)}} \\{\Delta\;{N(l)}}\end{pmatrix}}},{{{Z^{T}(l)}{h(l)}} = \begin{pmatrix}{{sHZ}(l)} \\{{sH}(l)}\end{pmatrix}},{{\sum\limits_{l = 1}^{4}\;{{s^{T}(l)}{h(l)}}} = {{{{sH}(L)} - {{sH}(R)}} = {\Delta\;{sH}}}},{{\sum\limits_{l = 1}^{4}\;{{s^{T}(l)}{s(l)}}} = {{{N(L)} + {N(R)}} = {N.}}}$Projection of 3-D coordinates to Laser Coordinates

The parameters a(i), b(i) specified as 8 parameters of the above 9-parameter fit are actually projections of the 3-D thread axisparameters a_(x), a_(y), b_(x), b_(y). A projection matrix, P, definesthe mapping from the 5-parameter fit to the 9-parameter fit. The anglesa_(i) are the angles of the laser beams with respect to the stage (x,y)axes. For example, laser-1 is incident at 22.5 degrees, laser-2 at 67.5degrees, laser-3 at 112.5 degrees, and laser-4 at 157.4 degrees.

$\begin{pmatrix}{a(1)} \\{b(1)} \\{a(2)} \\{b(2)} \\{a(3)} \\{b(3)} \\{a(4)} \\{b(4)} \\r\end{pmatrix} = {{P\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}} = {\begin{pmatrix}{- {\sin\left( \alpha_{1} \right)}} & {\cos\left( \alpha_{1} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin\left( \alpha_{1} \right)}} & {\cos\left( \alpha_{1} \right)} & 0 \\{- {\sin\left( \alpha_{2} \right)}} & {\cos\left( \alpha_{2} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin\left( \alpha_{2} \right)}} & {\cos\left( \alpha_{2} \right)} & 0 \\{- {\sin\left( \alpha_{3} \right)}} & {\cos\left( \alpha_{3} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin\left( \alpha_{3} \right)}} & {\cos\left( \alpha_{3} \right)} & 0 \\{- {\sin\left( \alpha_{4} \right)}} & {\cos\left( \alpha_{4} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin\left( \alpha_{4} \right)}} & {\cos\left( \alpha_{4} \right)} & 0 \\0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}}}$

Using the projection matrix we can reduce the previous matrix vectorequation from a 9-dimension problem to a 5-dimension problem.

${Z\begin{pmatrix}{a(1)} \\{b(1)} \\{a(2)} \\{b(2)} \\{a(3)} \\{b(3)} \\{a(4)} \\{b(4)} \\r\end{pmatrix}} = {{{ZP}\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}} = H}$

For just one laser, the equation is quite simple.

z_(i)(l, s)a(l) + b(l) + δ(s)r = h_(i)(l, s), where${l \equiv {{laser}\mspace{14mu}{index}}},{s \equiv {{sensor}\mspace{14mu}{index}}},{i \equiv {{data}\mspace{14mu}{index}}},{{a(l)} = {{{- {\sin\left( \alpha_{l} \right)}}a_{x}} + {{\cos\left( \alpha_{l} \right)}a_{y}}}},\begin{matrix}{{{b(l)} = {{{- \sin}\left( \alpha_{l} \right)b_{x}} + {{\cos\left( {\alpha\; l} \right)}b_{y}}}},{and}} \\{{\delta(s)} = {\left( {+ 1} \right)\mspace{14mu}{for}\mspace{14mu}\left( {s=={Left}} \right)}} \\{= {\left( {- 1} \right)\mspace{14mu}{for}\mspace{14mu}{\left( {s=={Right}} \right).}}}\end{matrix}$

This is the least squares solution:(a _(x) a _(y) b _(x) b _(y) r)^(T)=(P ^(T)(Z ^(T) Z)P)⁻¹(P ^(T) Z^(T))H.

The vector (P^(T)Z^(T))H can be computed in the simplified valuenotation. It is just the linear combination of sumHZ(l) or sumH(l) termsweighted by cos(α_(l)) or sin (α_(l)) factors.

${P^{T}\left( {Z^{T}H} \right)} = {\begin{pmatrix}{- {\sin\left( \alpha_{1} \right)}} & 0 & {- {\sin\left( \alpha_{2} \right)}} & 0 & {- {\sin\left( \alpha_{3} \right)}} & 0 & {- {\sin\left( \alpha_{4} \right)}} & 0 & 0 \\{\cos\left( \alpha_{1} \right)} & 0 & {\cos\left( \alpha_{2} \right)} & 0 & {\cos\left( \alpha_{3} \right)} & 0 & {\cos\left( \alpha_{4} \right)} & 0 & 0 \\0 & {- {\sin\left( \alpha_{1} \right)}} & 0 & {- {\sin\left( \alpha_{2} \right)}} & 0 & {- {\sin\left( \alpha_{3} \right)}} & 0 & {- {\sin\left( \alpha_{4} \right)}} & 0 \\0 & {\cos\left( \alpha_{1} \right)} & 0 & {\cos\left( \alpha_{2} \right)} & 0 & {\cos\left( \alpha_{3} \right)} & 0 & {\cos\left( \alpha_{4} \right)} & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}{{sHZ}(1)} \\{{sH}(1)} \\{{sHZ}(2)} \\{{sH}(2)} \\{{sHZ}(3)} \\{{sH}(3)} \\{{sHZ}(4)} \\{{sH}(4)} \\{\Delta\;{sH}}\end{pmatrix}}$ ${P^{T}\left( {Z^{T}H} \right)} = \begin{pmatrix}{- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{l} \right)}{{sHZ}(l)}}}} \\{+ {\sum\limits_{l = 1}^{4}\;{{\cos\left( \alpha_{l} \right)}{{sHZ}(l)}}}} \\{- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{l} \right)}{{sH}(l)}}}} \\{+ {\sum\limits_{l = 1}^{4}\;{{\cos\left( \alpha_{l} \right)}{{sH}(l)}}}} \\{\Delta\;{sH}}\end{pmatrix}$

The matrix (P^(T) Z^(T) ZP) also can be computed in the simplified valuenotation.

$\left( {{P^{T}\left( {Z^{T}Z} \right)}P} \right) = {{P^{T}\begin{pmatrix}{{sSqZ}(1)} & {{sZ}(1)} & 0 & 0 & 0 & 0 & 0 & 0 & {\Delta\;{{sZ}(1)}} \\{{sZ}(1)} & {N(1)} & 0 & 0 & 0 & 0 & 0 & 0 & {\Delta\;{N(1)}} \\0 & 0 & {{sSqZ}(2)} & {{sZ}(2)} & 0 & 0 & 0 & 0 & {\Delta\;{{sZ}(2)}} \\0 & 0 & {{sZ}(2)} & {N(2)} & 0 & 0 & 0 & 0 & {\Delta\;{N(2)}} \\0 & 0 & 0 & 0 & {{sSqZ}(3)} & {{sZ}(3)} & 0 & 0 & {\Delta\;{{sZ}(3)}} \\0 & 0 & 0 & 0 & {{sZ}(3)} & {N(3)} & 0 & 0 & {{\Delta\;{N(3)}}\;} \\0 & 0 & 0 & 0 & 0 & 0 & {{sSqZ}(4)} & {{sZ}(4)} & {\Delta\;{{sZ}(4)}} \\0 & 0 & 0 & 0 & 0 & 0 & {{sZ}(4)} & {N(4)} & {\Delta\;{N(4)}} \\{\Delta\;{{sZ}(1)}} & {\Delta\;{N(1)}} & {\Delta\;{{sZ}(2)}} & {\Delta\;{N(2)}} & {\Delta\;{{sZ}(3)}} & {\Delta\;{{sZ}(4)}} & {\Delta\;{{sZ}(4)}} & {\Delta\;{N(4)}} & N\end{pmatrix}}\begin{pmatrix}{- {\sin\left( \alpha_{1} \right)}} & {\cos\left( \alpha_{1} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin\left( \alpha_{1} \right)}} & {\cos\left( \alpha_{1} \right)} & 0 \\{- {\sin\left( \alpha_{2} \right)}} & {\cos\left( \alpha_{2} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin\left( \alpha_{2} \right)}} & {\cos\left( \alpha_{2} \right)} & 0 \\{- {\sin\left( \alpha_{3} \right)}} & {\cos\left( \alpha_{3} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin\left( \alpha_{3} \right)}} & {\cos\left( \alpha_{3} \right)} & 0 \\{- {\sin\left( \alpha_{4} \right)}} & {\cos\left( \alpha_{4} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin\left( \alpha_{4} \right)}} & {\cos\left( \alpha_{4} \right)} & 0 \\0 & 0 & 0 & 0 & 1\end{pmatrix}}$intermediate result:

$\left. {\left( {{P^{T}\left( {Z^{T}Z} \right)}P} \right) = P^{T}} \right)\left( \begin{matrix}{{- {\sin\left( \alpha_{1} \right)}}{{sSqZ}(1)}} & {{\cos\left( \alpha_{1} \right)}{{sSqZ}(1)}} & {{- {\sin\left( \alpha_{1} \right)}}{{sZ}(1)}} & {{\cos\left( \alpha_{1} \right)}{{sZ}(1)}} & {\Delta\;{{sZ}(1)}} \\{{- {\sin\left( \alpha_{1} \right)}}{{sZ}(1)}} & {{\cos\left( \alpha_{1} \right)}{{sZ}(1)}} & {{- {\sin\left( \alpha_{1} \right)}}{N(1)}} & {{\cos\left( \alpha_{1} \right)}{N(1)}} & {\Delta\;{N(1)}} \\{{- {\sin\left( \alpha_{2} \right)}}{{sSqZ}(2)}} & {{\cos\left( \alpha_{2} \right)}{{sSqZ}(2)}} & {{- {\sin\left( \alpha_{2} \right)}}{{sZ}(2)}} & {{\cos\left( \alpha_{2} \right)}{{sZ}(2)}} & {\Delta\;{{sZ}(2)}} \\{{- {\sin\left( \alpha_{2} \right)}}{{sZ}(2)}} & {{\cos\left( \alpha_{2} \right)}{{sZ}(2)}} & {{- {\sin\left( \alpha_{2} \right)}}{N(2)}} & {{\cos\left( \alpha_{2} \right)}{N(2)}} & {\Delta\;{N(2)}} \\{{- {\sin\left( \alpha_{3} \right)}}{{sSqZ}(3)}} & {{\cos\left( \alpha_{3} \right)}{{sSqZ}(3)}} & {{- {\sin\left( \alpha_{3} \right)}}{{sZ}(3)}} & {{\cos\left( \alpha_{3} \right)}{{sZ}(3)}} & {\Delta\;{{sZ}(3)}} \\{{- {\sin\left( \alpha_{3} \right)}}{{sZ}(3)}} & {{\cos\left( \alpha_{3} \right)}{{sZ}(3)}} & {{- {\sin\left( \alpha_{3} \right)}}{N(3)}} & {{\cos\left( \alpha_{3} \right)}{N(3)}} & {\Delta\;{N(3)}} \\{{- {\sin\left( \alpha_{4} \right)}}{{sSqZ}(4)}} & {{\cos\left( \alpha_{4} \right)}{{sSqZ}(4)}} & {{- {\sin\left( \alpha_{4} \right)}}{{sZ}(4)}} & {{\cos\left( \alpha_{4} \right)}{{sZ}(4)}} & {\Delta\;{{sZ}(4)}} \\{{- {\sin\left( \alpha_{4} \right)}}{{sZ}(4)}} & {{\cos\left( \alpha_{4} \right)}{{sZ}(4)}} & {{- {\sin\left( \alpha_{4} \right)}}{N(4)}} & {{\cos\left( \alpha_{4} \right)}{N(4)}} & {\Delta\;{N(4)}} \\{- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}\Delta\;{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\cos\left( \alpha_{1} \right)}\Delta\;{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}\Delta\;{N(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\cos\left( \alpha_{1} \right)}\Delta\;{N(l)}}}} & N\end{matrix} \right)$

The final result is a 5×5 matrix with the sin() and cos() terms mixingthe data from individual lasers.

$\left( {{P^{T}\left( {Z^{T}Z} \right)}P} \right) = \left( \begin{matrix}{+ {\sum\limits_{l = 1}^{4}\;{{\sin^{2}\left( \alpha_{1} \right)}{{sSqZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}{{sSqZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\sin^{2}\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}\Delta\;{{sZ}(l)}}}} \\{- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}{{sSqZ}(l)}}}} & {+ {\sum\limits_{i = 1}^{4}\;{{\cos^{2}\left( \alpha_{1} \right)}{{sSqZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\cos^{2}\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\cos\left( \alpha_{1} \right)}\Delta\;{{sZ}(l)}}}} \\{+ {\sum\limits_{l = 1}^{4}\;{{\sin^{4}\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\sin^{2}\left( \alpha_{1} \right)}{N(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}{N(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}\Delta\;{N(l)}}}} \\{- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\cos^{2}\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}{N(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\cos^{2}\left( \alpha_{1} \right)}{N(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\cos\left( \alpha_{1} \right)}\Delta\;{N(l)}}}} \\{- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}\Delta\;{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\cos\left( \alpha_{1} \right)}\Delta\;{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}\;{{\sin\left( \alpha_{1} \right)}\Delta\;{N(l)}}}} & {+ {\sum\limits_{l = 1}^{4}\;{{\cos\left( \alpha_{1} \right)}\Delta\;{N(l)}}}} & N\end{matrix} \right)$

One can define a set of 4-element vectors to make the previous resultslook more compact.

$\sin\limits^{\rightharpoonup} = \left( {{\sin\left( \alpha_{1} \right)}\mspace{20mu}{\sin\left( \alpha_{2} \right)}\mspace{20mu}{\sin\left( \alpha_{3} \right)}\mspace{20mu}{\sin\left( \alpha_{4} \right)}} \right)$$\cos\limits^{\rightharpoonup} = \left( {{\cos\left( \alpha_{1} \right)}\mspace{20mu}{\cos\left( \alpha_{2} \right)}\mspace{20mu}{\cos\left( \alpha_{3} \right)}\mspace{20mu}{\cos\left( \alpha_{4} \right)}} \right)$$\overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)} = \left( {{\sin\left( \alpha_{1} \right)}{\cos\left( \alpha_{1} \right)}\mspace{20mu}{\sin\left( \alpha_{2} \right)}{\cos\left( \alpha_{2} \right)}\mspace{20mu}{\sin\left( \alpha_{3} \right)}{\cos\left( \alpha_{3} \right)}\mspace{20mu}{\sin\left( \alpha_{4} \right)}{\cos\left( \alpha_{4} \right)}} \right)$$\overset{\rightharpoonup}{\left( \sin \right)^{2}} = \left( {{\sin^{2}\left( \alpha_{1} \right)}\mspace{20mu}{\sin^{2}\left( \alpha_{2} \right)}\mspace{20mu}{\sin^{2}\left( \alpha_{3} \right)}\mspace{20mu}{\sin^{2}\left( \alpha_{4} \right)}} \right)$$\overset{\rightharpoonup}{\left( \cos \right)^{2}} = \left( {{\cos^{2}\left( \alpha_{1} \right)}\mspace{20mu}{\cos^{2}\left( \alpha_{2}\; \right)}\mspace{20mu}{\cos^{2}\left( \alpha_{3} \right)}\mspace{20mu}{\cos^{2}\left( \alpha_{4} \right)}} \right)$$\overset{\rightharpoonup}{sSqZ} = \left( {{{sSqZ}(1)}\mspace{20mu}{{sSqZ}(2)}\mspace{20mu}{{sSqZ}(3)}\mspace{20mu}{{sSqZ}(4)}} \right)$$\overset{\rightharpoonup}{sZ} = \left( {{{sZ}(1)}\mspace{20mu}{{sZ}(2)}\mspace{20mu}{{sZ}(3)}\mspace{20mu}{{sZ}(4)}} \right)$$\overset{\rightharpoonup}{sH} = \left( {{{sH}(1)}\mspace{20mu}{{sH}(2)}\mspace{20mu}{{sH}(3)}\mspace{20mu}{{sH}(4)}} \right)$$\overset{\rightharpoonup}{sHZ} = \left( {{{sHZ}(1)}\mspace{20mu}{{sHZ}(2)}\mspace{20mu}{{sHZ}(3)}\mspace{20mu}{{sHZ}(4)}} \right)$$\overset{\rightharpoonup}{N} = \left( {{N(1)}\mspace{20mu}{N(2)}\mspace{20mu}{N(3)}\mspace{20mu}{N(4)}} \right)$$\overset{\rightharpoonup}{\Delta\; N} = \left( {\Delta\;{N(1)}\mspace{20mu}\Delta\;{N(2)}\mspace{20mu}\Delta\;{N(3)}\mspace{20mu}\Delta\;{N(4)}} \right)$Δ sH = (Δ sH(1)  Δ sH(2)  Δ sH(3)  Δ sH(4))Δ sZ = (Δ sZ(1)  Δ sZ(2)  Δ sZ(3)  Δ sZ(4))  

The final results are below.

Every term is the dot product of a geometrical vector representing theincident angles of each of the 4 lasers and a data vector representingwhat is observed in each of the 4 lasers. There are 15 independentnumbers to be calculated for (P^(T)(Z^(T) Z)P) and 5 for P^(T)(Z^(T) H).

${{\mathbb{P}}^{T}\left( {{\mathbb{Z}}^{T}{\mathbb{H}}} \right)} = \begin{pmatrix}{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sHZ}} \\{{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sHZ}} \\{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sH}} \\{{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sH}} \\{{+ \Delta}\;{sH}}\end{pmatrix}$$\left( {{{\mathbb{P}}^{T}\left( {{\mathbb{Z}}^{T}{\mathbb{Z}}} \right)}\;{\mathbb{P}}} \right) = \begin{pmatrix}{{+ \overset{\rightharpoonup}{\left( \sin \right)^{2}}} \cdot \overset{\rightharpoonup}{sSqZ}} & {{- \overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)}} \cdot \overset{\rightharpoonup}{sSqZ}} & {{+ \overset{\rightharpoonup}{\left( \sin \right)^{2}}} \cdot \overset{\rightharpoonup}{sZ}} & {{- \overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)}} \cdot \overset{\rightharpoonup}{sZ}} & {{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta\;{sZ}}} \\{{- \overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)}} \cdot \overset{\rightharpoonup}{sSqZ}} & {{+ \overset{\rightharpoonup}{\left( \cos \right)^{2}}} \cdot \overset{\rightharpoonup}{sSqZ}} & {{- \overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)}} \cdot \overset{\rightharpoonup}{sZ}} & {{+ \overset{\rightharpoonup}{\left( \cos \right)^{2}}} \cdot \overset{\rightharpoonup}{sZ}} & {{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta\;{sZ}}} \\{{+ \overset{\rightharpoonup}{\left( \sin \right)^{2}}} \cdot \overset{\rightharpoonup}{sZ}} & {{- \overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)}} \cdot \overset{\rightharpoonup}{sZ}} & {{+ \overset{\rightharpoonup}{\left( \sin \right)^{2}}} \cdot \overset{\rightharpoonup}{N}} & {{- \overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)}} \cdot \overset{\rightharpoonup}{N}} & {{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta\; N}} \\{{- \overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)}} \cdot \overset{\rightharpoonup}{sZ}} & {{+ \overset{\rightharpoonup}{\left( \cos \right)^{2}}} \cdot \overset{\rightharpoonup}{sZ}} & {{- \overset{\rightharpoonup}{\left( {\sin\mspace{11mu}\cos} \right)}} \cdot \overset{\rightharpoonup}{N}} & {{+ \overset{\rightharpoonup}{\left( \cos \right)^{2}}} \cdot \overset{\rightharpoonup}{N}} & {{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta\; N}} \\{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta\;{sZ}}} & {{+ \overset{\rightharpoonup}{\cos^{2}}} \cdot \overset{\rightharpoonup}{\Delta\;{sZ}}} & {{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta\; N}} & {{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta\; N}} & {+ N}\end{pmatrix}$Simple Cases

Suppose the cylinder is exactly aligned with the z-axis, and the data isexactly centered above and below each sensor's center line. Then in eachsensor, the left sensor will have a measurement points of type (x, h)and the right sensor will have measurements of type (x, −h). This willmean that the sHZ and sH terms will be zero. The term ΔsH will be a sumof +h and −(−h) terms or ΔsH≈Nh.

The solution equation would then read:

${\left( {{{\mathbb{P}}^{T}\left( {{\mathbb{Z}}^{T}{\mathbb{Z}}} \right)}{\mathbb{P}}} \right)\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}} = {{\begin{pmatrix}\cdots & \cdots & \cdots & \cdots & 0 \\\cdots & \cdots & \cdots & \cdots & 0 \\\cdots & \cdots & \cdots & \cdots & 0 \\\cdots & \cdots & \cdots & \cdots & 0 \\0 & 0 & 0 & 0 & N\end{pmatrix}\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}} = {{{\mathbb{P}}^{T}\left( {{\mathbb{Z}}^{T}{\mathbb{H}}} \right)} = {\begin{pmatrix}{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sHZ}} \\{{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sHZ}} \\{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sH}} \\{{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sH}} \\{{+ \Delta}\;{sH}}\end{pmatrix} = \begin{pmatrix}0 \\0 \\0 \\0 \\{Nh}\end{pmatrix}}}}$This has the solutiona _(x) =a _(y) =b _(x) =b _(y)=0, r=Nh/N=h.

Appendix D

The method of assembly and alignment of the optical and mechanicalcomponents of the light plane generator receiver and light planegenerator modules is executed utilizing the alignment fixture 100 ofFIG. 31. The alignment fixture includes an optical rail, modifiedbreadboards 120, 151 and 180, and a plurality of custom and commerciallyavailable opto-mechanical positioning devices. A detector assemblygenerally indicated at 159 and located on breadboard 151 replicates thefunction of a light plane receiver module.

The alignment fixture 100 further includes a references laser 118,aligned such that the center line of its light beam is horizontallyparallel to the reference breadboards 120 and 151 and lens and mirrormount 304 interfaces. Preferably, the laser 118 is a spatially filteredsolid state laser such as the 40001 available from LumenFlow Corp. ofMiddleville, Mich.

Also shown supported on the breadboard 120 is a reference prism (i.e.,rhomboid prism) assembly 122, the use of which is detailed herein. As itis the method of assembly that is important to proper functioningmodules, this disclosure emphasizes the interfacing components of thefixture while focusing on the detailed steps of the method to producethe modules.

Alignment fixture 100 of FIG. 31 includes a rail and a plurality ofstage assemblies, generally indicated at 102, which, in turn, includes arail assembly, generally indicated at 104. The rail assembly 104includes a rail 106 which is supported by pairs of bench legs 108. Thealignment fixture 100 also includes a set of rail carriages 110 slidablymounted on the rail 106.

The alignment fixture 100 further includes a transmitter stage assembly,generally indicated at 112. The transmitter stage assembly 112 includesan L1 (i.e., first lens) manipulator stage assembly, generally indicatedat 113, an L2 (i.e., second lens) manipulator stage assembly 114 and anL3 (i.e., third lens) manipulator stage assembly 116.

Each of the assemblies 113, 114 and 116 are supported together with thelaser 118 and its support bracket 119 on the breadboard or substrate120.

Also shown supported on the breadboard 120 is the reference prism (i.e.,rhomboid prism) assembly 122, an alignment aperture 124, a filter mount126 and a post holder 128.

The precision rhomboid prism assembly 122 is important to the assemblyof the modules, calibration of the align and focus instrument, and toassembly of the optical head.

The assembly 122 includes two metal plates, two dowel pins, and arhomboid prism.

The rhomboid prism assembly 122 is the gage that establishes the heightdifference between the height of laser light entering the transmitter(through lens 316) (preferably, 0.984″) and the height of laser lightexiting the transmitter (through lens 310) (preferably, 1.679″).

The height is set by:

-   -   precisely machined dowel pin positions; and    -   the precise shape of the rhomboid.

The rhomboid prism has the property that a straight beam of lightentering the prism exists the prism in a straight line and in a pathexactly parallel to the beam path of the entry beam. The heightdifference between the entry and exit beams is set by the rotationalangle of the prism, relative to the entry beam.

The alignment fixture 100 also includes a clamp post assembly 130 with akinematic ball also supported on the breadboard 120.

The alignment fixture 100 further includes a relay telescope assembly,generally indicated at 132, which, in turn, includes a relay doubletassembly 134 and an IR doublet 136. The relay telescope 132 is mountedon its carriage 110 by a scope mount 138.

Mounted on the rail 106 is another carriage 110 on which a post 140 issupported at one end of the fixture 100. The post 140 supports a filteror target holder 142. In turn, the holder 142 supports a target 210.

The alignment fixture 100 also includes a receiver stage assembly 150which, in turn, includes the breadboard or substrate 151. The receiverstage assembly 150 includes an alignment assembly 152 including analignment aperture 154.

The receiver stage assembly 150 further includes an optical rail 156supported on the breadboard 151. The detector assembly, generallyindicated at 159, is adjustably mounted on the rail 156. The detectorassembly 159 includes electronic boards 157, together with the sensormount with an aperture 155. The receiver stage assembly 150 furtherincludes a filter mount 160 and a vertical slit 162.

The receiver stage assembly 150 also includes a manipulator bracket 164and an assembly 166 having a kinematic ball 168 mounted at a distal endthereof.

The receiver stage assembly 150 further includes an L4 (i.e., fourthlens) manipulator stage assembly 170.

The L1 manipulator stage assembly 113 includes the breadboard 180 onwhich an x-y stage assembly 184, together with a kinematic base 186 aremounted. Adjustment screws 182 are provided to adjust the position ofthe stage assembly 184. Clamping arms 188 are mounted at a distal end ofthe stage assembly 113.

FIGS. 31 and 37 taken together illustrate: a clamp for lens 310 attachedto screw 116; a screw attached to clamp 114 below plate 120; arotational screw to rod attached to clamp 188; lower two screw holes of322 establish a rotational adjustment axis for 322; and upper one screwhole of 322 establishes control of rotation about 322 adjustment axis.

Overview of the Transmitter Alignment Process

The alignment of the transmitter module's optical components isimportant to the operation of the Laser Lab system. The alignment isaccomplished with the align and focus (i.e., A&F) instrument or fixture100.

The transmitter module is mounted in the A&F instrument 100, usingreference surfaces “A”, “B”, and “C” of the module. Reference surface“A” mounts flat to base plate 120, which has been aligned parallel tothe laser beam used in the A&F instrument 100, generated by the laser118. Reference surface “C” is made flush to two kinematic mounts thathave been aligned parallel to the laser beam 118. Reference surface “B”is flush to one kinematic mount and establishes the correct positionalong the beam line of the laser 118.

Mirror-2 is mounted to reference surface “E” on module 304. Mirror-1 ismounted to plate 322 which is mounted to reference surface “F”.

In the alignment process the optical components are configured to meetsystem requirements using the process previously detailed. During thealignment process the A&F instrument 100 holds the parts in place with aset of clamps and piece holders. When the alignment process iscompleted, the optical components are fixed to the surfaces of module304 with a glue that permanently holds them in place.

The following is a list of transmitter optical module components andtheir adjustments that are fixed in the align and focus instrument 100:

-   -   first cylindrical lens 316: plate 318 is clamped in A&F part        188. 188 can be moved in three directions: linearly parallel to        laser beam and linearly perpendicular to laser beam (with screws        182), and rotated about the axis of the rod holding clamp 188.    -   second cylindrical lens 312: linearly perpendicular to laser        beam with screw below plate 120, attached to 114. The lens can        also be rotated within clamp 114.    -   third cylindrical lens 310: linearly perpendicular to laser        beam, in up-down relation to plate 120, by screw 116. The lens        can also be rotated within clamp attached to screw 116.    -   adjustable mirror mount 322: radial adjustment about axis        established by lower two screws mounting 322, to 304. Adjustment        controlled by top screw of 322.

Thus, there are a total of eight independent adjustment parameters forthe transmitter module. Each parameter is optimized in the alignmentprocess, and fixed with glue before removal of the module 66 from theA&F instrument 100.

1.0 Produce light plane generator module assembly:

-   -   1.1 Install machined lens and mirror mount 304 to breadboard        120. Place an alignment aperture in the reference laser beam        path so that the laser beam passes through the aperture without        creating distortion.    -   1.2 Install lens 312 within the mount 304 and align so that the        beam passes through the aperture and is coincident with the        laser beam axis. Remove the aperture from the beam path.    -   1.3 Assemble L1 sub-assembly 320 components, cylinder lens 316,        L1 lens mount plate 318 and lens mount base 319. Assemble        sub-assembly 320 to mount 304.    -   1.4 Adjust rotational position of lens 312 about the laser beam        axis so that beam profile is level (FIG. 32).    -   1.5 Adjust rotational position of lens 316 about the one degree        of freedom provided so that the beam profile is flat (FIG. 33).    -   1.6 Adjust position of lens 316 along laser beam axis so that        beam width is visually equal at a distance greater than or equal        to 216″ and at 4″ distance from exit aperture of machined lens        and mirror mount 304.    -   1.7 Adjust the horizontal position of lens 316 so that power        distribution is visually balanced within the beam profile.    -   1.8 Re-adjust the horizontal position of lens 312 so that beam        is centered along the laser beam axis.    -   1.9 Re-adjust position of lens 316 along laser beam axis so that        full angle laser beam divergence is less than 0.25 m radians.        θ=(d2−d1)/(l2−l1)        where θ is beam divergence in radians; d1 is the horizontal beam        width at distance 1; d2 is the horizontal beam width at distance        2; l1 is the distance from the output aperture of machined lens        and mirror mount 304 to the beam measurement point at distance        1; l2 is the distance from the output aperture of machined lens        and mirror mount 304 to the beam measurement point at distance        2.    -   1.10 Assemble lens 310 to machined lens and mirror mount 304.        Ensure lens 310 is intimate with machined lens and mirror mount        304.    -   1.11 Place a 150 um slit aperture in laser beam path with slit        parallel to machined lens and mirror mount 304 mounting surface        at a height of 1.679″±0.002″. Adjust vertical position of lens        304 so that beam passes through slit opening. Alignment is        optimum when the maximum amount of light passes through        aperture. Remove slit aperture from beam path.    -   1.12 Adjust position of lens 310 rotationally so that beam        profile is not distorted (i.e., unlike FIG. 35) and generally        forms a rectangle (i.e., like FIG. 36).    -   1.13 Place the detector assembly 159 of the alignment fixture        100 in the laser beam path. Adjust lens 312 so that the        attenuated beam profile is aligned with the laser beam axis.    -   1.14 Adjust lens 316 position horizontally so that output power        is distributed equally ±5 mV across laser beam profile as        measured using detector assembly 159 and dual trace oscilloscope        readout.    -   1.15 Verify collimation, beam height, distortion, power        distribution and beam pointing. Correct as necessary. Apply and        cure adhesives to fixedly mount optical components to the mount        304.    -   1.16 Remove assembled light plane generator module from        alignment fixture 100.

2.0 Produce light plane receiver module assembly:

-   -   2.1 Install reference rhomboid prism 122 to breadboard 120 to        establish laser beam offset height parallel to machined lens and        mirror mount 500 of 1.679″.    -   2.2 Install machined lens and mirror mount 500 to breadboard        151.    -   2.3 Assemble lens 506 to machined lens and mirror mount 500.        Ensure lens 506 is intimate with machined lens and mirror mount        500.    -   2.4 Align lens 506 so that the beam is coincident with the laser        beam axis.    -   2.5 Place a cylinder lens 316 in the beam path utilizing the        cylinder lens holder provided within alignment fixture 100        positioned so that the laser plane output is parallel to the        mounting surface of machined lens and mirror mount 500.    -   2.6 Assemble L5 sub-assembly components, receiver spherical lens        508 (two) and lens mount plate 504 to machined lens and mirror        mount 500 by screws 516 (FIG. 47).    -   2.7 Place a target card with a line scribed at 1.679″±0.002″ at        back surface of machined lens and mirror mount 500.    -   2.8 Adjust lens pair vertically so that laser beam strikes        scribe line.    -   2.9 Adjust horizontal position of lens pair 508 so that laser        beam strikes pair center seam such that resulting spots are        visually equal in brightness. Remove target card.    -   2.10 Assemble aperture elements 510 and Hamamatsu photodetectors        514 to detector mount 502 using screws 505.    -   2.11 Assemble detector mount 502 to machined lens and mirror        mount 500 using screws 505.    -   2.12 Adjust position of detector mount 502 horizontally so that        beams are focused onto detectors 514 without clipping aperture        element 510 edges.    -   2.13 Remove rhomboid prism 122 and cylinder lens 316 from        fixture. Install a selected light plane generator module to        breadboard 120.    -   2.14 Adjust horizontal position of lens pair 508 so that output        power is distributed equally ±5 mV across laser beam profile as        measured using Hamamatsu photodetectors 514 output and dual        trace oscilloscope readout.    -   2.15 Verify that beams are focused onto detectors 514 without        clipping aperture element 510 edges.    -   2.16 Apply and cure adhesives to fixedly mount optical        components to the mount 500.    -   2.17 Remove assembled light plane generator module and light        plane receiver module pair from alignment fixture 100.

1. A method for precisely assembling an optical module, the methodcomprising: a) providing a set of optical components including aplurality of beam shaping components for converting at least one initialshape in cross section of a laser beam to at least one desired shape incross section and an optical mount for supporting the set of opticalcomponents; b) holding and locating the optical mount relative to areference axis; c) holding the optical components in position relativeto the optical mount during step b) wherein the positions of the beamshaping components are initial positions; d) directing a laser beamhaving the at least one initial shape at the set of optical componentsduring step c); e) adjusting the initial positions of the beam shapingcomponents to obtain final positions of the beam shaping componentsrelative to the optical mount during step d) until the laser beam hasthe at least one desired shape; and f) fixedly securing the opticalcomponents to the optical mount wherein the optical mount supports andis fixedly secured to the beam shaping components in their respectivefinal positions after step e) when the laser beam has the at least onedesired shape.
 2. The method as claimed in claim 1, wherein the at leastone initial shape is circular or nearly circular in cross section andthe at least one desired shape is generally rectangular in crosssection.
 3. The method as claimed in claim 2, wherein divergence of thelaser beam having the desired, generally rectangular shape is less thanor equal to a predetermined maximum value.
 4. The method as claimed inclaim 3, wherein the maximum value is 1.3 mrad or less.
 5. The method asclaimed in claim 2, wherein the beam shaping components include a firstlens held by the optical mount to move therewith during step c) andwherein the method further comprises the step of g) adjusting linearposition of the optical mount and the held first lens along thereference axis.
 6. The method as claimed in claim 5, further comprisingthe step of measuring width of the laser beam at a plurality ofdistances from the first lens during step d) to obtain a plurality ofwidth measurements.
 7. The method as claimed in claim 6, wherein thelaser beam has the desired, generally rectangular shape when the widthmeasurements are substantially equal.
 8. The method as claimed in claim5, wherein step e) includes the step of adjusting the angular positionof the first lens relative to the optical mount until a profile of thelaser beam is substantially flat.
 9. The method as claimed in claim 8,wherein step e) includes the step of adjusting the horizontal positionof the first lens relative to the optical mount until power distributionin the laser beam is balanced within the profile.
 10. The method asclaimed in claim 5, wherein the beam shaping components include a secondlens and wherein step e) includes the step of adjusting angular positionof the second lens relative to the optical mount until a profile of thelaser beam is level.
 11. The method as claimed in claim 10, wherein stepe) includes the step of adjusting horizontal position of the second lensrelative to the optical mount until the laser beam is centered along thereference axis.
 12. The method as claimed in claim 10, wherein the beamshaping components include a third lens and wherein step e) includes thestep of adjusting vertical position of the third lens relative to theoptical mount so that the laser beam is located in the same verticalposition at a plurality of distances from the third lens.
 13. The methodas claimed in claim 12, wherein step e) includes the step of adjustingangular position of the third lens relative to the optical mount. 14.The method as claimed in claim 2, wherein the desired, generallyrectangular shape has a substantially flat profile.
 15. The method asclaimed in claim 2, wherein the beam shaping components include a firstlens and wherein step e) includes the step of adjusting angular positionof the first lens relative to the optical mount until a profile of thelaser beam is substantially flat.
 16. A light plane generator moduleassembled by performing the steps of claim
 2. 17. The module as claimedin claim 16, wherein the beam shaping components include first, secondand third cylindrical lenses.
 18. A light plane receiver moduleassembled by performing the steps of claim 1 wherein the at least oneinitial shape is a pair of spaced, generally rectangular shapes incross-section and the at least one desired shape is a pair of spacedcircular shapes in cross-section.
 19. The module as claimed in claim 18,wherein the set of optical components includes a pair of spaced lensesand a corresponding pair of spaced photodetectors.
 20. The module asclaimed in claim 19, wherein the spaced lenses comprise a pair of spacedspherical lenses.
 21. The module as claimed in claim 20, wherein the setof optical components include a pair of spaced apertured elementsdisposed between the pair of spaced spherical lenses and the pair ofspaced photodetectors.